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Exponential distribution
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Not to be confused with the exponential family of probability distributions.
Exponential
Probability density function
Cumulative distribution function
Parameters
λ > 0 rate, or inverse scale
Support
x ∈ [0, ∞)
PDF
λ e−λx
CDF
1 − e−λx
Quantile
−ln(1 − F) / λ
Mean
λ−1 (= β)
Median
λ−1 ln(2)
Mode
0
Variance
λ−2 (= β2)
Skewness
2
Ex. kurtosis
6
Entropy
1 − ln(λ)
MGF
λ λ − t , for t < λ {\displaystyle {\frac {\lambda }{\lambda -t}},{\text{ for }}t<\lambda }
CF
λ λ − i t {\displaystyle {\frac {\lambda }{\lambda -it}}}
Fisher information
λ − 2 {\displaystyle \lambda ^{-2}}
In probability theory and statistics, the exponential distribution (also known as negative exponential distribution) is the probability distribution that describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson processes, it is found in various other contexts.
The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.
Contents
[hide]
1 Characterization
1.1 Probability density function
1.2 Cumulative distribution function
1.3 Alternative parameterization
2 Properties
2.1 Mean, variance, moments and median
2.2 Memorylessness
2.3 Quantiles
2.4 Kullback–Leibler divergence
2.5 Maximum entropy distribution
2.6 Distribution of the minimum of exponential random variables
2.7 Joint moments of i.i.d. exponential order statistics
3 Parameter estimation
3.1 Maximum likelihood
3.2 Approximate minimizer of expected squared error
3.3 Confidence intervals
3.4 Bayesian inference
4 Generating exponential variates
5 Related distributions
6 Applications of exponential distribution
6.1 Occurrence of events
6.2 Prediction
7 See also
8 References
9 External links
Characterization[edit]
Probability density function[edit]
The probability density function (pdf) of an exponential distribution is
f ( x ; λ ) = { λ e − λ x x ≥ 0 , 0 x < 0. {\displaystyle f(x;\lambda )={\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}
Alternatively, this can be defined using the right-continuous Heaviside step function, H(x) where H(0)=1:
f ( x ; λ ) = λ e − λ x H ( x ) {\displaystyle f(x;\lambda )=\mathrm {\lambda } e^{-\lambda x}H(x)}
Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0, ∞). If a random variable X has this distribution, we write X ~ Exp(λ).
The exponential distribution exhibits infinite divisibility.
Cumulative distribution function[edit]
The cumulative distribution function is given by
F ( x ; λ ) = { 1 − e − λ x x ≥ 0 , 0 x < 0 {\displaystyle F(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0\end{cases}}}
Alternatively, this can be defined using the Heaviside step function, H(x).
F ( x ; λ ) = ( 1 − e − λ x ) H ( x ) {\displaystyle F(x;\lambda )=\mathrm {(} 1-e^{-\lambda x})H(x)}
Alternative parameterization[edit]
A commonly used alternative parametrization is to define the probability density function (pdf) of an exponential distribution as
f ( x ; β ) = { 1 β e − x β x ≥ 0 , 0 x < 0. {\displaystyle f(x;\beta )={\begin{cases}{\frac {1}{\beta }}e^{-{\frac {x}{\beta }}}&x\geq 0,\\0&x<0.\end{cases}}}
where β > 0 is mean, standard deviation, and scale parameter of the distribution, the reciprocal of the rate parameter, λ, defined above. In this specification, β is a survival parameter in the sense that if a random variable X is the duration of time that a given biological or mechanical system manages to survive and X ~ Exp(β) then E[X] = β. That is to say, the expected duration of survival of the system is β units of time. The parametrization involving the "rate" parameter arises in the context of events arriving at a rate λ, when the time between events (which might be modeled using an exponential distribution) has a mean of β = λ−1.
The alternative specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. This alternative specification is not used here. Unfortunately this gives rise to a notational ambiguity. In general, the reader must check which of these two specifications is being used if an author writes "X ~ Exp(λ)", since either the notation in the previous (using λ) or the notation in this section (here, using β to avoid confusion) could be intended. An example of this notational switch: reference[1] uses λ for β.
Properties[edit]
Mean, variance, moments and median[edit]
The mean is the probability mass centre, that is the first moment.
The median is the preimage F−1(1/2).
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by
E  [ X ] = 1 λ = β {\displaystyle \operatorname {E} [X]={\frac {1}{\lambda }}=\beta } , see above.
In light of the examples given above, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call.
The variance of X is given by
Var  [ X ] = 1 λ 2 = β 2 {\displaystyle \operatorname {Var} [X]={\frac {1}{\lambda ^{2}}}=\beta ^{2}} ,
so the standard deviation is equal to the mean.
The moments of X, for n = 1, 2, ..., are given by
E  [ X n ] = n ! λ n {\displaystyle \operatorname {E} \left[X^{n}\right]={\frac {n!}{\lambda ^{n}}}} .
The median of X is given by
m  [ X ] = ln  ( 2 ) λ < E  [ X ] {\displaystyle \operatorname {m} [X]={\frac {\ln(2)}{\lambda }}<\operatorname {E} [X]} ,
where ln refers to the natural logarithm. Thus the absolute difference between the mean and median is
| E  [ X ] − m  [ X ] | = 1 − ln  ( 2 ) λ < 1 λ = standard deviation {\displaystyle |\operatorname {E} [X]-\operatorname {m} [X]|={\frac {1-\ln(2)}{\lambda }}<{\frac {1}{\lambda }}={\text{standard deviation}}} ,
in accordance with the median-mean inequality.
Memorylessness[edit]
An exponentially distributed random variable T obeys the relation
Pr ( T > s + t | T > s ) = Pr ( T > t ) , ∀ s , t ≥ 0 {\displaystyle \Pr \left(T>s+t|T>s\right)=\Pr(T>t),\qquad \forall s,t\geq 0} .
When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the conditional probability that occurrence will take at least 10 more seconds is equal to the unconditional probability of observing the event more than 10 seconds relative to the initial time.
The exponential distribution and the geometric distribution are the only memoryless probability distributions.
The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant Failure rate.
Quantiles[edit]
Tukey criteria for anomalies.[citation needed]
The quantile function (inverse cumulative distribution function) for Exp(λ) is
F − 1 ( p ; λ ) = − ln  ( 1 − p ) λ , 0 ≤ p < 1 {\displaystyle F^{-1}(p;\lambda )={\frac {-\ln(1-p)}{\lambda }},\qquad 0\leq p<1}
The quartiles are therefore:
first quartile: ln(4/3)/λ
median: ln(2)/λ
third quartile: ln(4)/λ
And as a consequence the interquartile range is ln(3)/λ.
Kullback–Leibler divergence[edit]
The directed Kullback–Leibler divergence of e λ {\displaystyle e^{\lambda }} ('approximating' distribution) from e λ 0 {\displaystyle e^{\lambda _{0}}} ('true' distribution) is given by
Δ ( λ 0 | | λ ) = log  ( λ 0 ) − log  ( λ ) + λ λ 0 − 1. {\displaystyle \Delta (\lambda _{0}||\lambda )=\log(\lambda _{0})-\log(\lambda )+{\frac {\lambda }{\lambda _{0}}}-1.}
Maximum entropy distribution[edit]
Among all continuous probability distributions with support [0, ∞) and mean μ, the exponential distribution with λ = 1/μ has the largest differential entropy. In other words, it is the maximum entropy probability distribution for a random variate X which is greater than or equal to zero and for which E[X] is fixed.[2]
Distribution of the minimum of exponential random variables[edit]
Let X1, ..., Xn be independent exponentially distributed random variables with rate parameters λ1, ..., λn. Then
min { X 1 , … , X n } {\displaystyle \min \left\{X_{1},\dotsc ,X_{n}\right\}}
is also exponentially distributed, with parameter
λ = λ 1 + ⋯ + λ n {\displaystyle \lambda =\lambda _{1}+\dotsb +\lambda _{n}} .
This can be seen by considering the complementary cumulative distribution function:
Pr ( min { X 1 , … , X n } > x ) = Pr ( X 1 > x , … , X n > x ) = ∏ i = 1 n Pr ( X i > x ) = ∏ i = 1 n exp  ( − x λ i ) = exp  ( − x ∑ i = 1 n λ i ) . {\displaystyle {\begin{aligned}\Pr \left(\min\{X_{1},\dotsc ,X_{n}\}>x\right)&=\Pr \left(X_{1}>x,\dotsc ,X_{n}>x\right)\\&=\prod _{i=1}^{n}\Pr(X_{i}>x)\\&=\prod _{i=1}^{n}\exp(-x\lambda _{i})=\exp \left(-x\sum _{i=1}^{n}\lambda _{i}\right).\end{aligned}}}
The index of the variable which achieves the minimum is distributed according to the law
Pr ( k | X k = min { X 1 , … , X n } ) = λ k λ 1 + ⋯ + λ n . {\displaystyle \Pr \left(k|X_{k}=\min\{X_{1},\dotsc ,X_{n}\}\right)={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.}
Note that
max { X 1 , … , X n } {\displaystyle \max\{X_{1},\dotsc ,X_{n}\}}
is not exponentially distributed.[3]
Joint moments of i.i.d. exponential order statistics[edit]
Let X 1 , … , X n {\displaystyle X_{1},\dotsc ,X_{n}} be n {\displaystyle n} independent and identically distributed exponential random variables with rate parameter λ. Let X ( 1 ) , … , X ( n ) {\displaystyle X_{(1)},\dotsc ,X_{(n)}} denote the corresponding order statistics. For i < j {\displaystyle i<j} , the joint moment E [ X ( i ) X ( j ) ] {\displaystyle E[X_{(i)}X_{(j)}]} of the order statistics X ( i ) {\displaystyle X_{(i)}} and X ( j ) {\displaystyle X_{(j)}} is given by
E [ X ( i ) X ( j ) ] = 1 ( n − j + 1 ) λ E [ X ( i ) ] + E [ X ( i ) 2 ] = 1 ( n − j + 1 ) λ 1 ( n − i + 1 ) λ + 2 ( ( n − i + 1 ) λ ) 2 . {\displaystyle E[X_{(i)}X_{(j)}]={\frac {1}{(n-j+1)\lambda }}E[X_{(i)}]+E[X_{(i)}^{2}]={\frac {1}{(n-j+1)\lambda }}{\frac {1}{(n-i+1)\lambda }}+{\frac {2}{((n-i+1)\lambda )^{2}}}.}
This can be seen by invoking the Law of total expectation and the memoryless property:
E [ X ( i ) X ( j ) ] = ∫ x = 0 ∞ E [ X ( i ) X ( j ) | X ( i ) = x ] f X ( i ) ( x ) d x = ∫ x = 0 ∞ x E [ X ( j ) | X ( j ) ≥ x ] f X ( i ) ( x ) d x ( since X ( i ) = x ⟹ X ( j ) ≥ x ) = ∫ x = 0 ∞ x [ 1 ( n − j + 1 ) λ + x ] f X ( i ) ( x ) d x ( by the memoryless property ) = 1 ( n − j + 1 ) λ E [ X ( i ) ] + E [ X ( i ) 2 ] . {\displaystyle {\begin{aligned}E[X_{(i)}X_{(j)}]&=\int _{x=0}^{\infty }E[X_{(i)}X_{(j)}|X_{(i)}=x]f_{X_{(i)}}(x)dx\\&=\int _{x=0}^{\infty }xE[X_{(j)}|X_{(j)}\geq x]f_{X_{(i)}}(x)dx\qquad ({\textrm {since}}~X_{(i)}=x\implies X_{(j)}\geq x)\\&=\int _{x=0}^{\infty }x\left[{\frac {1}{(n-j+1)\lambda }}+x\right]f_{X_{(i)}}(x)dx\qquad ({\text{by the memoryless property}})\\&={\frac {1}{(n-j+1)\lambda }}E[X_{(i)}]+E[X_{(i)}^{2}].\end{aligned}}}
The first equation follows from the Law of total expectation. The second equation exploits the fact that once we condition on X ( i ) = x {\displaystyle X_{(i)}=x} , it must follow that X ( j ) ≥ x {\displaystyle X_{(j)}\geq x} . The third equation relies on the memoryless property to replace E [ X ( j ) | X ( j ) ≥ x ] {\displaystyle E[X_{(j)}|X_{(j)}\geq x]} with E [ X ( j ) ] + x {\displaystyle E[X_{(j)}]+x} .
Parameter estimation[edit]
Below, suppose random variable X is exponentially distributed with rate parameter λ, and x 1 , … , x n {\displaystyle x_{1},\dotsc ,x_{n}} are n independent samples from X, with sample mean x ¯ {\displaystyle {\bar {x}}} .
Maximum likelihood[edit]
The likelihood function for λ, given an independent and identically distributed sample x = (x1, ..., xn) drawn from the variable, is:
L ( λ ) = ∏ i = 1 n λ exp  ( − λ x i ) = λ n exp  ( − λ ∑ i = 1 n x i ) = λ n exp  ( − λ n x ¯ ) , {\displaystyle L(\lambda )=\prod _{i=1}^{n}\lambda \exp(-\lambda x_{i})=\lambda ^{n}\exp \left(-\lambda \sum _{i=1}^{n}x_{i}\right)=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right),}
where:
x ¯ = 1 n ∑ i = 1 n x i {\displaystyle {\overline {x}}={1 \over n}\sum _{i=1}^{n}x_{i}}
is the sample mean.
The derivative of the likelihood function's logarithm is:
d d λ ln  ( L ( λ ) ) = d d λ ( n ln  ( λ ) − λ n x ¯ ) = n λ − n x ¯ { > 0 , 0 < λ < 1 x ¯ , = 0 , λ = 1 x ¯ , < 0 , λ > 1 x ¯ . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\ln(L(\lambda ))={\frac {\mathrm {d} }{\mathrm {d} \lambda }}\left(n\ln(\lambda )-\lambda n{\overline {x}}\right)={\frac {n}{\lambda }}-n{\overline {x}}\ {\begin{cases}>0,&0<\lambda <{\frac {1}{\overline {x}}},\\[8pt]=0,&\lambda ={\frac {1}{\overline {x}}},\\[8pt]<0,&\lambda >{\frac {1}{\overline {x}}}.\end{cases}}}
Consequently, the maximum likelihood estimate for the rate parameter is:
λ ^ = 1 x ¯ = n ∑ i x i {\displaystyle {\widehat {\lambda }}={\frac {1}{\overline {x}}}={\frac {n}{\sum _{i}x_{i}}}}
Although this is not an unbiased estimator of λ {\displaystyle \lambda } , x ¯ {\displaystyle {\overline {x}}} is an unbiased[4] MLE[5] estimator of 1 / λ = β , {\displaystyle 1/\lambda =\beta ,} where β {\displaystyle \beta } is the scale parameter defined in the 'Alternative parameterization' section above and the distribution mean.
Approximate minimizer of expected squared error[edit]
Assume you have at least three samples. If we seek a minimizer of expected mean squared error (see also: Bias–variance tradeoff) that is similar to the maximum likelihood estimate (i.e. a multiplicative correction to the likelihood estimate) we have:
λ ^ = ( n − 2 n ) ( 1 x ¯ ) = n − 2 ∑ i x i {\displaystyle {\widehat {\lambda }}=\left({\frac {n-2}{n}}\right)\left({\frac {1}{\bar {x}}}\right)={\frac {n-2}{\sum _{i}x_{i}}}}
This is derived from the mean and variance of the Inverse-gamma distribution: Inv-Gamma ( n , λ ) {\textstyle {\mbox{Inv-Gamma}}(n,\lambda )} .[6]
Confidence intervals[edit]
The 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by:[7]
2 n λ ^ χ 1 − α 2 , 2 n 2 < 1 λ < 2 n λ ^ χ α 2 , 2 n 2 {\displaystyle {\frac {2n}{{\widehat {\lambda }}\chi _{1-{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n}{{\widehat {\lambda }}\chi _{{\frac {\alpha }{2}},2n}^{2}}}}
which is also equal to:
2 n x ¯ χ 1 − α 2 , 2 n 2 < 1 λ < 2 n x ¯ χ α 2 , 2 n 2 {\displaystyle {\frac {2n{\overline {x}}}{\chi _{1-{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n{\overline {x}}}{\chi _{{\frac {\alpha }{2}},2n}^{2}}}}
where χ2
p,v is the 100(p) percentile of the chi squared distribution with v degrees of freedom, n is the number of observations of inter-arrival times in the sample, and x-bar is the sample average. A simple approximation to the exact interval endpoints can be derived using a normal approximation to the χ2
p,v distribution. This approximation gives the following values for a 95% confidence interval:
λ l o w = λ ^ ( 1 − 1.96 n ) {\displaystyle \lambda _{low}={\widehat {\lambda }}\left(1-{\frac {1.96}{\sqrt {n}}}\right)}
λ u p p = λ ^ ( 1 + 1.96 n ) {\displaystyle \lambda _{upp}={\widehat {\lambda }}\left(1+{\frac {1.96}{\sqrt {n}}}\right)}
This approximation may be acceptable for samples containing at least 15 to 20 elements.[8]
Bayesian inference[edit]
The conjugate prior for the exponential distribution is the gamma distribution (of which the exponential distribution is a special case). The following parameterization of the gamma probability density function is useful:
G a m m a ( λ ; α , β ) = β α Γ ( α ) λ α − 1 exp  ( − λ β ) . {\displaystyle \mathrm {Gamma} (\lambda ;\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\lambda ^{\alpha -1}\exp(-\lambda \beta ).}
The posterior distribution p can then be expressed in terms of the likelihood function defined above and a gamma prior:
p ( λ ) ∝ L ( λ ) × G a m m a ( λ ; α , β ) = λ n exp  ( − λ n x ¯ ) × β α Γ ( α ) λ α − 1 exp  ( − λ β ) ∝ λ ( α + n ) − 1 exp  ( − λ ( β + n x ¯ ) ) . {\displaystyle {\begin{aligned}p(\lambda )&\propto L(\lambda )\times \mathrm {Gamma} (\lambda ;\alpha ,\beta )\\&=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right)\times {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\lambda ^{\alpha -1}\exp(-\lambda \beta )\\&\propto \lambda ^{(\alpha +n)-1}\exp(-\lambda \left(\beta +n{\overline {x}}\right)).\end{aligned}}}
Now the posterior density p has been specified up to a missing normalizing constant. Since it has the form of a gamma pdf, this can easily be filled in, and one obtains:
p ( λ ) = G a m m a ( λ ; α + n , β + n x ¯ ) . {\displaystyle p(\lambda )=\mathrm {Gamma} (\lambda ;\alpha +n,\beta +n{\overline {x}}).}
Here the hyperparameter α can be interpreted as the number of prior observations, and β as the sum of the prior observations. The posterior mean here is:
α + n β + n x ¯ . {\displaystyle {\frac {\alpha +n}{\beta +n{\overline {x}}}}.}
Generating exponential variates[edit]
A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval (0, 1), the variate
T = F − 1 ( U ) {\displaystyle T=F^{-1}(U)}
has an exponential distribution, where F −1 is the quantile function, defined by
F − 1 ( p ) = − ln  ( 1 − p ) λ . {\displaystyle F^{-1}(p)={\frac {-\ln(1-p)}{\lambda }}.}
Moreover, if U is uniform on (0, 1), then so is 1 − U. This means one can generate exponential variates as follows:
T = − ln  ( U ) λ . {\displaystyle T={\frac {-\ln(U)}{\lambda }}.}
Other methods for generating exponential variates are discussed by Knuth[9] and Devroye.[10]
A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.[10]
Related distributions[edit]
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Exponential distribution is closed under scaling by a positive factor. If X ~ Exp(λ) then kX ~ Exp(λ/k).
If X ~ Exp(1/2) then X ∼ χ2
2, i.e. X has a chi-squared distribution with 2 degrees of freedom. Hence:
Exp ( λ ) = 1 2 λ Exp ( 1 2 ) ∼ 1 2 λ χ 2 2 ⇒ ∑ i = 1 n Exp ( λ ) ∼ 1 2 λ χ 2 n 2 {\displaystyle {\mbox{Exp}}\left(\lambda \right)={\frac {1}{2\lambda }}{\mbox{Exp}}\left({\frac {1}{2}}\right)\sim {\frac {1}{2\lambda }}\chi _{2}^{2}\Rightarrow \sum \limits _{i=1}^{n}{{\mbox{Exp}}\left(\lambda \right)}\sim {\frac {1}{2\lambda }}\chi _{2n}^{2}}
If X ~ Exp(λ) and Y ~ Exp(ν) then min(X, Y) ~ Exp(λ + ν).
If Xi ~ Exp(λ) then min{X1, ..., Xn} ~ Exp(nλ).
The Benktander Weibull distribution reduces to a truncated exponential distribution. If X ~ Exp(λ) then 1+X ~ BenktanderWeibull(λ, 1).
The exponential distribution is a limit of a scaled beta distribution:
lim n → ∞ n B e t a ( 1 , n ) = E x p ( 1 ) . {\displaystyle \lim _{n\to \infty }n{\rm {Beta}}(1,n)={\rm {Exp}}(1).}
If Xi ~ Exp(λ) then the sum X1 + ... + Xk = ∑ i X i {\displaystyle \sum _{i}X_{i}} ~ Erlang(k, λ) which is just a Gamma(k, λ−1) (in (k, θ) parametrization) or Gamma(k, λ) (in (α,β) parametrization) with an integer shape parameter k.
If X ~ Exp(1) then μ − σ log(X) ~ GEV(μ, σ, 0).
If X ~ Exp(λ) then X ~ Gamma(1, λ−1) (in (k, θ) parametrization) or Gamma(1, λ) (in (α, β) parametrization).
If X ~ Exp(λ) and Y ~ Exp(ν) then λX − νY ~ Laplace(0, 1).
If X, Y ~ Exp(λ) then X − Y ~ Laplace(0, λ−1).
If X ~ Laplace(μ, β−1) then |X − μ| ~ Exp(β).
If X ~ Exp(1) then (logistic distribution):
μ − β log  ( e − X 1 − e − X ) ∼ L o g i s t i c ( μ , β ) {\displaystyle \mu -\beta \log \left({\tfrac {e^{-X}}{1-e^{-X}}}\right)\sim \mathrm {Logistic} (\mu ,\beta )}
If X, Y ~ Exp(1) then (logistic distribution):
μ − β log  ( X Y ) ∼ L o g i s t i c ( μ , β ) {\displaystyle \mu -\beta \log \left({\tfrac {X}{Y}}\right)\sim \mathrm {Logistic} (\mu ,\beta )}
If X ~ Exp(λ) then keX ~ Pareto(k, λ).
If X ~ Pareto(1, λ) then log(X) ~ Exp(λ).
Exponential distribution is a special case of type 3 Pearson distribution.
If X ~ Exp(λ) then e−X ~ Beta(λ, 1).
If X ~ Exp(λ) then e − X k ∼ P o w e r L a w ( k , λ ) {\displaystyle {\tfrac {e^{-X}}{k}}\sim \mathrm {PowerLaw} (k,\lambda )} (power law)
If X ~ Exp(λ) then X {\displaystyle {\sqrt {X}}} ~ R a y l e i g h ( 1 / 2 λ ) {\displaystyle \mathrm {Rayleigh} \left(1/{\sqrt {2\lambda }}\right)} .
If X ~ Exp(λ) then X ∼ W e i b u l l ( 1 λ , 1 ) {\displaystyle X\sim \mathrm {Weibull} ({\tfrac {1}{\lambda }},1)} (Weibull distribution)
If Xi ~ U(0, 1) then
lim n → ∞ n min ( X 1 , … , X n ) ∼ Exp ( 1 ) {\displaystyle \lim _{n\to \infty }n\min \left(X_{1},\ldots ,X_{n}\right)\sim {\textrm {Exp}}(1)}
If Y|X ~ Poisson(X) where X ~ Exp(λ−1) then Y ∼ G e o m e t r i c ( 1 1 + λ ) {\displaystyle Y\sim \mathrm {Geometric} ({\tfrac {1}{1+\lambda }})} (geometric distribution)
If X ~ Exp(1) and Y ∼ Γ ( α , β α ) {\displaystyle Y\sim \Gamma (\alpha ,{\tfrac {\beta }{\alpha }})} then X Y ∼ K ( α , β ) {\displaystyle {\sqrt {XY}}\sim \mathrm {K} (\alpha ,\beta )} (K-distribution)
The Hoyt distribution can be obtained from Exponential distribution and Arcsine distribution
If X ~ Exp(λ) and Y ~ Erlang(n, λ) then:
X Y + 1 ∼ P a r e t o ( 1 , n ) {\displaystyle {\frac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}
If X ~ Exp(λ) and Y ∼ Γ ( n , 1 λ ) {\displaystyle Y\sim \Gamma (n,{\tfrac {1}{\lambda }})} then X Y + 1 ∼ P a r e t o ( 1 , n ) {\displaystyle {\tfrac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}
If X ~ SkewLogistic(θ), then log(1 + e−X) ~ Exp(θ).
If X ~ Exp(λ) and Y = μ − β log(Xλ) then Y ∼ Gumbel(μ, β).
Let X ∼ Exp(λX) and Y ∼ Exp(λY) be independent. Then Z = λ X X λ Y Y {\displaystyle Z={\frac {\lambda _{X}X}{\lambda _{Y}Y}}} has probability density function f Z ( z ) = 1 ( z + 1 ) 2 {\displaystyle f_{Z}(z)={\frac {1}{(z+1)^{2}}}} . This can be used to obtain a confidence interval for λ X λ Y {\displaystyle {\frac {\lambda _{X}}{\lambda _{Y}}}} .
Gamma mixture: If λ ~ Gamma(scale=k, shape=θ) and X ~ Exponential(rate=λ) then the marginal distribution of X is Lomax(scale=1/k, shape=θ)
Other related distributions:
Hyper-exponential distribution – the distribution whose density is a weighted sum of exponential densities.
Hypoexponential distribution – the distribution of a general sum of exponential random variables.
exGaussian distribution – the sum of an exponential distribution and a normal distribution.
Applications of exponential distribution[edit]
Occurrence of events[edit]
The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous Poisson process.
The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.
In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. Similar caveats apply to the following examples which yield approximately exponentially distributed variables:
The time until a radioactive particle decays, or the time between clicks of a geiger counter
The time it takes before your next telephone call
The time until default (on payment to company debt holders) in reduced form credit risk modeling
Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand, or between roadkills on a given road.
In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. (The arrival of customers for instance is also modeled by the Poisson distribution if the arrivals are independent and distributed identically.) The length of a process that can be thought of as a sequence of several independent tasks follows the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).
Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using CumFreq[11]
Reliability theory and reliability engineering also make extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constant hazard rate portion of the bathtub curve used in reliability theory. It is also very convenient because it is so easy to add failure rates in a reliability model. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the "failure rates" here are not constant: more failures occur for very young and for very old systems.
In physics, if you observe a gas at a fixed temperature and pressure in a uniform gravitational field, the heights of the various molecules also follow an approximate exponential distribution, known as the Barometric formula. This is a consequence of the entropy property mentioned below.
In hydrology, the exponential distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.[12]
The blue picture illustrates an example of fitting the exponential distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
Prediction[edit]
Having observed a sample of n data points from an unknown exponential distribution a common task is to use these samples to make predictions about future data from the same source. A common predictive distribution over future samples is the so-called plug-in distribution, formed by plugging a suitable estimate for the rate parameter λ into the exponential density function. A common choice of estimate is the one provided by the principle of maximum likelihood, and using this yields the predictive density over a future sample xn+1, conditioned on the observed samples x = (x1, ..., xn) given by
p M L ( x n + 1 ∣ x 1 , … , x n ) = ( 1 x ¯ ) exp  ( − x n + 1 x ¯ ) {\displaystyle p_{\rm {ML}}(x_{n+1}\mid x_{1},\ldots ,x_{n})=\left({\frac {1}{\overline {x}}}\right)\exp \left(-{\frac {x_{n+1}}{\overline {x}}}\right)}
The Bayesian approach provides a predictive distribution which takes into account the uncertainty of the estimated parameter, although this may depend crucially on the choice of prior.
A predictive distribution free of the issues of choosing priors that arise under the subjective Bayesian approach is
p C N M L ( x n + 1 ∣ x 1 , … , x n ) = n n + 1 ( x ¯ ) n ( n x ¯ + x n + 1 ) n + 1 , {\displaystyle p_{\rm {CNML}}(x_{n+1}\mid x_{1},\ldots ,x_{n})={\frac {n^{n+1}\left({\overline {x}}\right)^{n}}{\left(n{\overline {x}}+x_{n+1}\right)^{n+1}}},}
which can be considered as
(1) a frequentist confidence distribution, obtained from the distribution of the pivotal quantity x n + 1 / x ¯ {\displaystyle {x_{n+1}}/{\overline {x}}} ;[13]
(2) a profile predictive likelihood, obtained by eliminating the parameter λ from the joint likelihood of xn+1 and λ by maximization;[14]
(3) an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior 1/λ;
(4) the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations.[15]
The accuracy of a predictive distribution may be measured using the distance or divergence between the true exponential distribution with rate parameter, λ0, and the predictive distribution based on the sample x. The Kullback–Leibler divergence is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(λ0||p) denote the Kullback–Leibler divergence between an exponential with rate parameter λ0 and a predictive distribution p it can be shown that
E λ 0 [ Δ ( λ 0 ∣∣ p M L ) ] = ψ ( n ) + 1 n − 1 − log  ( n ) E λ 0 [ Δ ( λ 0 ∣∣ p C N M L ) ] = ψ ( n ) + 1 n − log  ( n ) {\displaystyle {\begin{aligned}{\rm {E}}_{\lambda _{0}}\left[\Delta (\lambda _{0}\mid \mid p_{\rm {ML}})\right]&=\psi (n)+{\frac {1}{n-1}}-\log(n)\\{\rm {E}}_{\lambda _{0}}\left[\Delta (\lambda _{0}\mid \mid p_{\rm {CNML}})\right]&=\psi (n)+{\frac {1}{n}}-\log(n)\end{aligned}}}
where the expectation is taken with respect to the exponential distribution with rate parameter λ0 ∈ (0, ∞), and ψ( · ) is the digamma function. It is clear that the CNML predictive distribution is strictly superior to the maximum likelihood plug-in distribution in terms of average Kullback–Leibler divergence for all sample sizes n > 0.
See also[edit]
Dead time – an application of exponential distribution to particle detector analysis.
Laplace distribution, or the "double exponential distribution".
Relationships among probability distributions
Marshall–Olkin exponential distribution
References[edit]
Jump up ^ David Olive, Chapter 4. Truncated Distributions, "Lemma 4.3", Southern Illinois University, February 18, 2010, p.107.
Jump up ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. Elsevier: 219–230. Retrieved 2011-06-02.
Jump up ^ Michael, Lugo. "The expectation of the maximum of exponentials" (PDF). Retrieved 13 December 2016.
Jump up ^ Richard Arnold Johnson; Dean W. Wichern (2007). Applied Multivariate Statistical Analysis. Pearson Prentice Hall. ISBN 978-0-13-187715-3. Retrieved 10 August 2012.
Jump up ^ NIST/SEMATECH e-Handbook of Statistical Methods
Jump up ^ Abdulaziz Elfessi and David M. Reineke, "A Bayesian Look at Classical Estimation: The Exponential Distribution", Journal of Statistics Education Volume 9, Number 1 (2001).
Jump up ^ Ross, Sheldon M. (2009). Introduction to probability and statistics for engineers and scientists (4th ed.). Associated Press. p. 267. ISBN 978-0-12-370483-2.
Jump up ^ Guerriero, V. (2012). "Power Law Distribution: Method of Multi-scale Inferential Statistics". Journal of Modern Mathematics Frontier (JMMF). 1: 21–28.
Jump up ^ Donald E. Knuth (1998). The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn. Boston: Addison–Wesley. ISBN 0-201-89684-2. See section 3.4.1, p. 133.
^ Jump up to: a b Luc Devroye (1986). Non-Uniform Random Variate Generation. New York: Springer-Verlag. ISBN 0-387-96305-7. See chapter IX, section 2, pp. 392–401.
Jump up ^ "Cumfreq, a free computer program for cumulative frequency analysis".
Jump up ^ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis (PDF). Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN 90-70754-33-9. CS1 maint: Extra text: authors list (link)
Jump up ^ Lawless, J.F., Fredette, M.,"Frequentist predictions intervals and predictive distributions", Biometrika (2005), Vol 92, Issue 3, pp 529–542.
Jump up ^ Bjornstad, J.F. (1990). "Predictive Likelihood: A Review". Statist. Sci. 5 (2): 242–254. doi:10.1214/ss/1177012175.
Jump up ^ D. F. Schmidt and E. Makalic, "Universal Models for the Exponential Distribution", IEEE Transactions on Information Theory, Volume 55, Number 7, pp. 3087–3090, 2009 doi:10.1109/TIT.2009.2018331
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Exponential distribution", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Online calculator of Exponential Distribution
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Probability density function

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Cumulative distribution function

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values: \operatorname{E}\left [X^n \right ] = \frac{n!}{\lambda^n}

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values: \operatorname{m}[X] = \frac{\ln(2)}{\lambda} < \operatorname{E}[X]

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values: |\operatorname{E}[X]- \operatorname{m}[X]| = \frac{1- \ln(2)}{\lambda}< \frac{1}{\lambda} = \text{standard deviation}

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values: \Pr \left (T > s + t | T > s \right ) = \Pr(T > t), \qquad \forall s, t \ge 0

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values: \Delta(\lambda_0 || \lambda) = \log(\lambda_0) - \log(\lambda) + \frac{\lambda}{\lambda_0} - 1.

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Independent random variables

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 \begin{align}
\Pr \left (\min\{X_1,\dotsc,X_n \} > x \right ) & = \Pr\left(X_1 > x, \dotsc, X_n > x\right) \\
&= \prod_{i=1}^n \Pr(X_i > x) \\
&= \prod_{i=1}^n \exp(-x\lambda_i) = \exp\left(-x\sum_{i=1}^n \lambda_i\right).
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values: Independent identically-distributed random variables

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values: L(\lambda) = \prod_{i=1}^n \lambda \exp(-\lambda x_i) = \lambda^n \exp \left(-\lambda \sum_{i=1}^n x_i\right)=\lambda^n\exp\left(-\lambda n \overline{x}\right),

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values: {\frac  {{\mathrm  {d}}}{{\mathrm  {d}}\lambda }}\ln(L(\lambda ))={\frac  {{\mathrm  {d}}}{{\mathrm  {d}}\lambda }}\left(n\ln(\lambda )-\lambda n\overline {x}\right)={\frac  {n}{\lambda }}-n\overline {x}\ {\begin{cases}>0,&0<\lambda <{\frac  {1}{\overline {x}}},\\[8pt]=0,&\lambda ={\frac  {1}{\overline {x}}},\\[8pt]<0,&\lambda >{\frac  {1}{\overline {x}}}.\end{cases}}

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Maximum likelihood

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Unbiased estimator

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values: \frac{2n}{\widehat{\lambda} \chi^2_{1-\frac{\alpha}{2},2n}} < \frac{1}{\lambda} < \frac{2n}{\widehat{\lambda} \chi^2_{\frac{\alpha}{2},2n}}

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values: \frac{2n\overline{x}}{\chi^2_{1-\frac{\alpha}{2},2n}} < \frac{1}{\lambda} < \frac{2n\overline{x}}{\chi^2_{\frac{\alpha}{2},2n}}

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Chi squared distribution

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p(\lambda) &\propto L(\lambda) \times \mathrm{Gamma}(\lambda; \alpha, \beta) \\
&= \lambda^n \exp\left (-\lambda n\overline{x} \right) \times \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda \beta) \\
&\propto \lambda^{(\alpha+n)-1} \exp(-\lambda \left (\beta + n\overline{x} \right)).
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Inverse transform sampling method

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Benktander Weibull distribution

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Skew-logistic distribution

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ExGaussian distribution

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Queuing theory

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Reliability theory

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Hazard rate

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Confidence belt

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Plotting position

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values: p_{\rm ML}(x_{n+1} \mid x_1, \ldots, x_n) = \left( \frac1{\overline{x}} \right) \exp \left( - \frac{x_{n+1}}{\overline{x}} \right)

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values: p_{\rm CNML}(x_{n+1} \mid x_1, \ldots, x_n) = \frac{ n^{n+1} \left( \overline{x} \right)^n }{ \left( n \overline{x} + x_{n+1} \right)^{n+1} },

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\begin{align}
{\rm E}_{\lambda_0} \left[ \Delta(\lambda_0\mid\mid p_{\rm ML}) \right] &= \psi(n) + \frac{1}{n-1} - \log(n) \\
{\rm E}_{\lambda_0} \left[ \Delta(\lambda_0\mid\mid p_{\rm CNML}) \right] &= \psi(n) + \frac{1}{n} - \log(n)
\end{align}

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text:
Escrow
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
This article is about an escrow of money. For escrows of computer source code, see source code escrow.
It has been suggested that Stakeholder (law) be merged into this article. (Discuss) Proposed since September 2017.
An escrow is a contractual arrangement in which a third party receives and disburses money or documents for the primary transacting parties, with the disbursement dependent on conditions agreed to by the transacting parties, or an account established by a broker for holding funds on behalf of the broker's principal or some other person until the consummation or termination of a transaction;[1] or, a trust account held in the borrower's name to pay obligations such as property taxes and insurance premiums. The word derives from the Old French word escroue, meaning a scrap of paper or a scroll of parchment; this indicated the deed that a third party held until a transaction was completed.[2]
Contents
1 Types
1.1 Internet escrow
1.2 Banking
1.3 Intellectual property
1.4 Law
1.5 Real estate
1.6 Mergers and acquisitions
2 Legal implications
3 See also
4 References
5 External links
Types[edit]
Escrow generally refers to money held by a third-party on behalf of transacting parties. It is mostly used regarding the purchase of shares of a company. It is best known in the United States in the context of real estate (specifically in mortgages where the mortgage company establishes an escrow account to pay property tax and insurance during the term of the mortgage).[3][unreliable source?] Escrow is an account separate from the mortgage account where deposit of funds occurs for payment of certain conditions that apply to the mortgage, usually property taxes and insurance. The escrow agent has the duty to properly account for the escrow funds and ensure that usage of funds is explicitly for the purpose intended. Since a mortgage lender is not willing to take the risk that a homeowner will not pay property tax, escrow is usually required under the mortgage terms. Escrow companies are also commonly used in the transfer of high value personal and business property, like websites and businesses, and in the completion of person-to-person remote auctions (such as eBay), although the advent of new low cost online escrow services has meant that even low cost transactions are now starting to benefit from use of escrow. In the UK, escrow accounts are often used during private property transactions to hold solicitors' clients' money, such as the deposit, until such time as the transaction completes.[4] Other examples include purchases of a second hand car, where the money may be released at the end of a warranty period, deposits for a property rental, where the money is released after the tenant moves out, provision of construction services, where the money may be released when the building work is complete to a defined standard, or when defined parts of the work are complete, perishable food products, sold on live seafood auctions (such as Gfresh).
An unrelated type of escrow is when a purchaser of a complex system, such as bespoke process control software or a large industrial installation, may require the supplier to place the design into source code escrow, so that the purchaser remains in a position to maintain and modify the system in case of the demise of the supplier.
Internet escrow[edit]
Internet escrow has been around since the beginning of Internet auctions and commerce. It was one of the many developments that allowed for trust to be established in the online sphere.[5]
As with traditional escrow, Internet escrow works by placing money in the control of an independent and licensed third party in order to protect both buyer and seller in a transaction. When both parties verify the transaction has been completed per terms set, the money is released. If at any point there is a dispute between the parties in the transaction, the process moves along to dispute resolution. The outcome of the dispute resolution process will decide what happens to money in escrow. With the growth of both business and individual commerce on the web traditional escrow companies have been supplanted by new technologies.
In the United States, the California Department of Business Oversight enacted Internet escrow companies as a licensed class effective 1 July 2001.[6] The first Internet escrow company to be licensed was Escrow.com,[7] founded by Fidelity National Financial in 1999.[8]
In the European Union, the Payment Services Directive, which commenced on 1 November 2009, has for the first time allowed the introduction of very low-cost Internet escrow services that are properly licensed and government-regulated. The regulatory framework in the EU allows these web-based escrow services, which operate along the lines of expensive Letter of Credit service run by banks for international buyers and sellers, but at a cost in cents rather than thousands of Euros, the ability to enhance security in commercial transactions.[9]
Bogus escrow methods have been employed online. In an effort to persuade a wary Internet auction participant, the perpetrator will propose the use of a third-party escrow service. The victim is unaware that the perpetrator has actually created an escrow site that closely resembles a legitimate escrow service. The victim sends payment to the phony escrow company and receives nothing in return. Alternatively, the victim sends merchandise to the subject and waits for his/her payment through the escrow site, which is never received because it is illegitimate.[10] Genuine online escrow companies will be listed on a government register, and it is advised that users should never use an online escrow service without first verifying that it is genuine by independently viewing a government on-line register. Currently there is not a license offered by the US Federal Government for online escrow services. However, certain states offer their own license for online escrow services; such as the California Department of Business[11] and the Arizona Department of Financial Institutions.[12]
Banking[edit]
Escrow is used in the field of automatic banking and vending equipment. One example is automated teller machines (ATMs), and is the function which allows the machine to hold the money deposited by the customer separately, and in case he or she challenges the counting result, the money is returned. Another example is a vending machine, where the customer's money is held in a separate escrow area pending successful completion of the transaction. If a problem occurs and the customer presses the refund button, the coins are returned from escrow; if no problem occurs, they fall into the coin vault of the machine.[13]
Intellectual property[edit]
Source code escrow agents hold source code of software in escrow just as other escrow companies hold cash. Normally[dubious – discuss] you do not own or have any rights to the software (including source code) that you are accessing, under the terms of a regular SaaS or desktop software agreement. This does not usually become an issue until technical problems start to arise, i.e. unexpected service interruptions, downtime, loss of application functionality and loss of data. This can add significant costs to your business and you remain reliant upon the software supplier to resolve these issues, unless you have an escrow agreement in place. Escrow is when the software source code is held by a third party – an escrow agent – on behalf of the customer and the supplier.[citation needed] Information escrow agents, such as the International Creative Registry, hold in escrow intellectual property and other information. Examples include song music and lyrics, manufacturing designs and laboratory notebooks, and television and movie treatments and scripts. This is done to establish legal ownership rights, with the independent escrow agents attesting to the information's ownership, contents, and creation date.
Law[edit]
Escrow is also known in the judicial context. So-called escrow funds are commonly used to distribute money from a cash settlement in a class action or environmental enforcement action. This way the defendant is not responsible for distribution of judgment moneys to the individual plaintiffs or the court-determined use (such as environmental remediation or mitigation). The defendant pays the total amount of the judgment (or settlement) to the court-administered or appointed escrow fund, and the fund distributes the money (often reimbursing its expenses from the judgment funds).
Real estate[edit]
In the U.S., escrow payment is a common term referring to the portion of a mortgage payment that is designated to pay for real property taxes and hazard insurance. It is an amount "over and above" the principal and interest portion of a mortgage payment. Since the escrow payment is used to pay taxes and insurance, it is referred to as "T&I", while the mortgage payment consisting of principal and interest is called "P&I". The sum total of all elements is then referred to as "PITI", for "Principal, Interest, Tax, and Insurance". Some mortgage companies require customers to maintain an escrow account that pays the property taxes and hazard insurance. Others offer it as an option for customers. Some types of loans, most notably Federal Housing Administration (FHA) loans, require the lender to maintain an escrow account for the life of the loan.
Even with a fixed interest rate, monthly mortgage payments may change over the life of the loan due to changes in property taxes and insurance premiums. For instance, if a hazard insurance premium increases by $120.00 per year, the escrow payment will need to increase by $10.00 per month to account for this difference (in addition to collection for the resulting escrow shortage when the mortgage company paid $120.00 more for the hazard insurance premium than what was anticipated). By RESPA guidelines the escrow payment must be recomputed at least once every 12 months to account for increases in property taxes or insurance. This is called an escrow analysis.
The escrow payment used to pay T&I is a long term escrow account that may last for years or for the life of the loan. Escrow can also refer to a shorter term account used to facilitate the closing of a real estate transaction. In this type of escrow, the escrow company holds onto all documents and money related to closing the transaction, rather than having the buyer and the seller deal directly with each other. When and if the transaction is ready to close, the escrow company distributes all funds and documents to their rightful recipients, and records the deed with the appropriate authorities.[14]
The validity of an online escrow company's license can be checked with the regulating authority. Usually this is accessible through their official website, and should always be checked before dealing with an escrow company to ensure it is not a look-alike or fraud.
This service is offered by multiple bankers as an outsourced product.
Mergers and acquisitions[edit]
Escrow arrangements are often used as part of mergers and acquisitions to supplement the warranties and indemnities offered by the seller(s).[15] This will be particularly likely where the credit risk of the seller(s) is of a poor quality and therefore the buyer is concerned about their ability to recover any sums that may become due.
Unlike many other forms of escrow, escrow arrangements in corporate transactions are often designed to last for extended periods rather than simply to complete the transfer of an asset. There is also commonly the requirement for an escrow agent to adjudicate on the validity of a claim on the escrow funds, which can lead to the risk of dispute between the parties.
Due to the length that the funds are held, the escrow arrangements need to take into account different considerations to those for other escrow arrangements, for example (i) information provision to the parties; (ii) application of interest earned on the funds; and (iii) credit worthiness of the financial institution.
Legal implications[edit]
Not all escrow agreements impose the duties of a legal trustee on the escrow agent, and in many such agreements, escrow agents are held to a mere gross negligence standard and benefit from indemnity and hold harmless provisions.
If the escrow agent is licensed by governmental authority[where?], then much higher legal standards may apply.
See also[edit]
Closing (real estate)
Commingling
Custodian bank
Delivery versus payment
Double escrow
Power of attorney
Real estate
Stewardship
Source code escrow
References[edit]
Jump up ^ "Escrow Process Elements & Tips". Archived from the original on 2007-09-16.
Jump up ^ "escrow - Origin and meaning of escrow by Online Etymology Dictionary". www.etymonline.com.
Jump up ^ "Escrow".
Jump up ^ UK, eSources. "Using Escrow For Payments (Payments Series - Article 2) - eSources.co.uk". articles.esources.co.uk.
Jump up ^ "eBay Community : Letter". pages.ebay.com.
Jump up ^ "California Financial Code, Division 6 Escrow Agents, Chapter 1, Section 17003". California Legislative Information. Retrieved 14 July 2016.
Jump up ^ "California Department of Corporations Online Escrow Fraud Questions & Answers" (PDF). California Department of Business Oversight. California Department of Corporations. Archived from the original (PDF) on 21 September 2016. Retrieved 14 July 2016.
Jump up ^ "Fidelity National Financial, Inc. to Offer Internet Escrow Services Through Escrow.com". PR Newswire Association LLC. Retrieved 14 July 2016 – via The Free Library.
Jump up ^ "Directive on Payment Services (PSD)". Archived from the original on 2014-11-05.
Jump up ^ "Internet Crime Complaint Center (IC3) - Internet Crime Schemes". www.ic3.gov.
Jump up ^ "California Department of Business".
Jump up ^ "Arizona Department of Financial Institutions".
Jump up ^ Up and Running, Tim Sanford, Vending Times, August 2005.
Jump up ^ "How Does an Escrow Account Work?". Time.com. Retrieved 9 June 2018.
Jump up ^ "Guide to M&A Escrow - Elemental CoSec". www.elementalcosec.com.
External links[edit]
Look up escrow in Wiktionary, the free dictionary.
"FAQs About Escrow Accounts for Consumers". U.S. Department of Housing and Urban Development. 2005-06-20. Archived from the original on 2007-04-03. Retrieved 2007-04-12.
"Escrow recommended for online and Bitcoin transactions". U.S. Department of Housing and Urban Development. 2013-06-20. Archived from the original on 2006-06-30.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Escrow&oldid=846088912"
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text:
Arable land
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Modern mechanized agriculture permits large fields like this one in Dorset, England.
Arable land (from Latin arabilis, "able to be plowed") is, according to one definition, land capable of being ploughed and used to grow crops.[1] In Britain, it was traditionally contrasted with pasturable lands such as heaths which could be used for sheep-rearing but not farmland.
A quite different kind of definition is used by various agencies concerned with agriculture. In providing statistics on arable land, the FAO and the World Bank[2] use the definition offered in the glossary accompanying FAOSTAT: "Arable land is the land under temporary agricultural crops (multiple-cropped areas are counted only once), temporary meadows for mowing or pasture, land under market and kitchen gardens and land temporarily fallow (less than five years). The abandoned land resulting from shifting cultivation is not included in this category. Data for ‘Arable land’ are not meant to indicate the amount of land that is potentially cultivable."[3] A briefer definition appearing in the Eurostat glossary similarly refers to actual, rather than potential use: "land worked (ploughed or tilled) regularly, generally under a system of crop rotation."[4]
Contents
[hide]
1 Arable land area
2 Arable land (hectares per person)
3 Non-arable land
4 See also
5 References
6 External links
Arable land area[edit]
World map of arable land, percentage by country (2006)[5]
According to Food and Agriculture Organization of the United Nations the world's Arable land amounted to 1,407 M ha, out of a total 4,924 M ha land used for agriculture, as for year 2013.[6]
Arable land area ('000 km2)[7][8]
Rank
Country or region
2008
2009
2010
2011
2012
— World 13,866 13,873 13,880 13,962 13,958
1 India 1,579 1,578 1,575 1,574 1,562
2 United States 1,631 1,605 1,598 1,602 1,551
3 Russia 1,216 1,218 1,200 1,215 1,197
— European Union 1,091 1,089 1,074 1,074 1,083
4 China 1,086 1,100 1,114 1,116 1,065
5 Brazil 702 704 703 719 726
6 Australia 440 475 426 477 471
7 Canada 443 438 434 430 459
8 Argentina 351 338 372 380 392
9 Nigeria 370 340 360 360 350
10 Ukraine 325 325 325 325 325
Arable land (hectares per person)[edit]
Fields in the region of Záhorie in Western Slovakia.
A field of sunflowers in Cardejón, Spain
Arable land (hectares per person)[7]
Country Name
2013
Afghanistan 0.254
Albania 0.213
Algeria 0.196
American Samoa 0.054
Andorra 0.038
Angola 0.209
Antigua and Barbuda 0.044
Argentina 0.933
Armenia 0.150
Aruba 0.019
Australia 1.999
Austria 0.160
Azerbaijan 0.204
Bahamas, The 0.021
Bahrain 0.001
Bangladesh 0.049
Barbados 0.039
Belarus 0.589
Belgium 0.073
Belize 0.227
Benin 0.262
Bermuda 0.005
Bhutan 0.133
Bolivia 0.427
Bosnia and Herzegovina 0.264
Botswana 0.125
Brazil 0.372
British Virgin Islands 0.034
Brunei Darussalam 0.012
Bulgaria 0.479
Burkina Faso 0.363
Burundi 0.115
Cabo Verde 0.108
Cambodia 0.275
Cameroon 0.279
Canada 1.306
Cayman Islands 0.003
Central African Republic 0.382
Chad 0.373
Channel Islands 0.026
Chile 0.074
China 0.078
Colombia 0.036
Comoros 0.086
Congo, Dem. Rep. 0.098
Congo, Rep. 0.125
Costa Rica 0.049
Côte d'Ivoire 0.134
Croatia 0.206
Cuba 0.278
Curaçao
Cyprus 0.070
Czech Republic 0.299
Denmark 0.429
Djibouti 0.002
Dominica 0.083
Dominican Republic 0.078
Ecuador 0.076
Egypt, Arab Rep. 0.031
El Salvador 0.120
Equatorial Guinea 0.151
Eritrea
Estonia 0.480
Ethiopia 0.160
Faroe Islands 0.062
Fiji 0.187
Finland 0.409
France 0.277
French Polynesia 0.009
Gabon 0.197
Gambia, The 0.236
Georgia 0.119
Germany 0.145
Ghana 0.180
Gibraltar
Greece 0.232
Greenland 0.016
Grenada 0.028
Guam 0.006
Guatemala 0.064
Guinea 0.259
Guinea-Bissau 0.171
Guyana 0.552
Haiti 0.103
Honduras 0.130
Hong Kong SAR, China 0.000
Hungary 0.445
Iceland 0.374
India 0.123
Indonesia 0.094
Iran, Islamic Rep. 0.193
Iraq 0.147
Ireland 0.242
Isle of Man 0.253
Israel 0.035
Italy 0.113
Jamaica 0.044
Japan 0.033
Jordan 0.032
Kazakhstan 1.726
Kenya 0.133
Kiribati 0.018
Korea, Dem. People’s Rep. 0.094
Korea, Rep. 0.030
Kosovo
Kuwait 0.003
Kyrgyz Republic 0.223
Lao PDR 0.226
Latvia 0.600
Lebanon 0.025
Lesotho 0.119
Liberia 0.116
Libya 0.274
Liechtenstein 0.070
Lithuania 0.774
Luxembourg 0.115
Macao SAR, China
Macedonia, FYR 0.199
Madagascar 0.153
Malawi 0.235
Malaysia 0.032
Maldives 0.010
Mali 0.386
Malta 0.021
Marshall Islands 0.038
Mauritania 0.116
Mauritius 0.060
Mexico 0.186
Micronesia, Fed. Sts. 0.019
Moldova 0.510
Monaco
Mongolia 0.198
Montenegro 0.013
Morocco 0.240
Mozambique 0.213
Myanmar 0.203
Namibia 0.341
Nauru
Nepal 0.076
Netherlands 0.062
New Caledonia 0.024
New Zealand 0.123
Nicaragua 0.253
Niger 0.866
Nigeria 0.197
Northern Mariana Islands 0.019
Norway 0.159
Oman 0.010
Pakistan 0.168
Palau 0.048
Panama 0.148
Papua New Guinea 0.041
Paraguay 0.696
Peru 0.136
Philippines 0.057
Poland 0.284
Portugal 0.107
Puerto Rico 0.017
Qatar 0.007
Romania 0.438
Russian Federation 0.852
Rwanda 0.107
Samoa 0.042
San Marino 0.032
São Tomé and Príncipe 0.048
Saudi Arabia 0.102
Senegal 0.229
Serbia 0.460
Seychelles 0.001
Sierra Leone 0.256
Singapore 0.000
Sint Maarten (Dutch part)
Slovak Republic 0.258
Slovenia 0.085
Solomon Islands 0.036
Somalia 0.107
South Africa 0.235
South Sudan
Spain 0.270
Sri Lanka 0.063
St. Kitts and Nevis 0.092
St. Lucia 0.016
St. Martin (French part)
St. Vincent and the Grenadines 0.046
Sudan 0.345
Suriname 0.112
Swaziland 0.140
Sweden 0.270
Switzerland 0.050
Syrian Arab Republic 0.241
Tajikistan 0.106
Tanzania 0.269
Thailand 0.249
Timor-Leste 0.131
Togo 0.382
Tonga 0.152
Trinidad and Tobago 0.019
Tunisia 0.262
Turkey 0.270
Turkmenistan 0.370
Turks and Caicos Islands 0.030
Tuvalu
Uganda 0.189
Ukraine 0.715
United Arab Emirates 0.004
United Kingdom 0.098
United States 0.480
Uruguay 0.682
Uzbekistan 0.145
Vanuatu 0.079
Venezuela, RB 0.089
Vietnam 0.071
Virgin Islands (U.S.) 0.010
West Bank and Gaza 0.011
Yemen, Rep. 0.049
Zambia 0.243
Zimbabwe 0.268
Non-arable land[edit]
This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2014) (Learn how and when to remove this template message)
Water Buffalo ploughing rice fields near Salatiga, Central Java, Indonesia
A pasture in the East Riding of Yorkshire in England
Agricultural land that is not arable according to the FAO definition above includes:
Permanent crop - land that produces crops from woody vegetation, e.g. orchardland, vineyards, coffee plantations, rubber plantations, and land producing nut trees;
Meadows and pastures - land used as pasture and grazed range, and those natural grasslands and sedge meadows that are used for hay production in some regions.
Other non-arable land includes land that is not suitable for any agricultural use.
Land that is not arable, in the sense of lacking capability or suitability for cultivation for crop production, has one or more limitations e.g. lack of sufficient fresh water for irrigation, stoniness, steepness, adverse climate, excessive wetness with impracticality of drainage, excessive salts, among others.[9] Although such limitations may preclude cultivation, and some will in some cases preclude any agricultural use, large areas unsuitable for cultivation are agriculturally productive. For example, US NRCS statistics indicate that about 59 percent of US non-federal pasture and unforested rangeland is unsuitable for cultivation, yet such land has value for grazing of livestock.[10] In British Columbia, Canada, 41 percent of the provincial Agricultural Land Reserve area is unsuitable for production of cultivated crops, but is suitable for uncultivated production of forage usable by grazing livestock.[11] Similar examples can be found in many rangeland areas elsewhere.
Land incapable of being cultivated for production of crops can sometimes be converted to arable land. New arable land makes more food, and can reduce starvation. This outcome also makes a country more self-sufficient and politically independent, because food importation is reduced. Making non-arable land arable often involves digging new irrigation canals and new wells, aqueducts, desalination plants, planting trees for shade in the desert, hydroponics, fertilizer, nitrogen fertilizer, pesticides, reverse osmosis water processors, PET film insulation or other insulation against heat and cold, digging ditches and hills for protection against the wind, and greenhouses with internal light and heat for protection against the cold outside and to provide light in cloudy areas. This process is often extremely expensive. An alternative is the Seawater Greenhouse which desalinates water through evaporation and condensation using solar energy as the only energy input. This technology is optimized to grow crops on desert land close to the sea.
(Note: The use of artifices does not make land arable. Rock, still remains rock, and shallow less than 6 feet turnable soil is still considered NONE toilable (IE: None arable). The use of artifice is an open air none recycled water hydroponics relationship. The below described circumstances are not in perspective, have limited duration, and have a tendency to accumulate trace materials in soil that either there or elsewhere cause de-oxination. IE: Use of fast amounts of fertilizer in the United States that end up devastating rivers, water ways and river endings due accumulation of none degradable toxins and Nitrogen bearing molecules that remove oxygen and cause none aerobic processes to form.)
Some examples of infertile non-arable land being turned into fertile arable land are:
Aran Islands: These islands off the west coast of Ireland, (not to be confused with the Isle of Arran in Scotland's Firth of Clyde), were unsuitable for arable farming because they were too rocky. The people covered the islands with a shallow layer of seaweed and sand from the ocean. Today, crops are grown there, even though, the islands are still considered none arable.
Israel: The construction of desalination plants along Israel's coast allowed agriculture in some areas that were formerly desert. The desalination plants, which remove the salt from ocean water, have created a new source of water for farming, drinking, and washing.
Slash and burn agriculture uses nutrients in wood ash, but these expire within a few years.
Terra preta, fertile tropical soils created by adding charcoal.
Some examples of fertile arable land being turned into infertile land are:
Droughts like the 'dust bowl' of the Great Depression in the U.S. turned farmland into desert.
Rainforest deforestation: The fertile tropical forests are converted into infertile desert land. For example, Madagascar's central highland plateau has become virtually totally barren (about ten percent of the country), as a result of slash-and-burn deforestation, an element of shifting cultivation practiced by many natives.
Each year, arable land is lost due to desertification and human-induced erosion. Improper irrigation of farm land can wick the sodium, calcium, and magnesium from the soil and water to the surface. This process steadily concentrates salt in the root zone, decreasing productivity for crops that are not salt-tolerant.
See also[edit]
Soil fertility
Land use statistics by country
List of environment topics
References[edit]
Jump up ^ Oxford English Dictionary, 3rd ed. "arable, adj. and n." Oxford University Press (Oxford), 2013.
Jump up ^ The World Bank. Agricultural land (% of land area) http://data.worldbank.org/indicator/AG.LND.AGRI.ZS
Jump up ^ FAOSTAT. [Statistical database of the Food and Agriculture Organization of the United Nations] Glossary. http://faostat3.fao.org/mes/glossary/E
Jump up ^ Eurostat. Glossary: Arable land. http://ec.europa.eu/eurostat/statistics-explained/index.php/Glossary:Arable_land
Jump up ^ Arable land in this map refers to a definition used by the US CIA - land cultivated for crops like wheat, maize, and rice that are replanted after each harvest
Jump up ^ "FAOSTAT Land Use module". Food and Agriculture Organization. Retrieved 8 July 2016.
^ Jump up to: a b "FAOSTAT Land Use module". Food and Agriculture Organization. Retrieved 8 July 2016.
Jump up ^ "Arable Land Area". The Helgi Library. Retrieved 22 February 2014.
Jump up ^ United States Department of Agriculture, Soil Conservation Service. 1961. Land capability classification. Agriculture Handbook 210. 21 pp.
Jump up ^ NRCS. 2013. Summary report 2010 national resources inventory. United States Natural Resources Conservation Service. 163 pp.
Jump up ^ Agricultural Land Commission. Agriculture Capability and the ALR Fact Sheet. http://www.alc.gov.bc.ca/alc/DownloadAsset?assetId=72876D8604EC45279B8D3C1B14428CF8&filename=agriculture_capability__the_alr_fact_sheet_2013.pdf
External links[edit]
Surface area of the Earth
Article from Technorati on Shrinking Arable Farmland in the world
Arable land: statistics by country
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