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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−1ln(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 point 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 point 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 Fisher Information
3.4 Confidence intervals
3.5 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 ∈ N {\displaystyle n\in \mathbb {N} } are given by
E  [ X n ] = n ! λ n {\displaystyle \operatorname {E} \left[X^{n}\right]={\frac {n!}{\lambda ^{n}}}} .
The central moments of X, for n ∈ N {\displaystyle n\in \mathbb {N} } are given by
μ n = ! n λ n = n ! λ n ∑ k = 0 n ( − 1 ) k k ! {\displaystyle \mu _{n}={\frac {!n}{\lambda ^{n}}}={\frac {n!}{\lambda ^{n}}}\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}} .
where !n is the subfactorial of 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} .
This can be seen by considering the complementary cumulative distribution function:
Pr ( T > s + t | T > s ) = Pr ( T > s + t ∩ T > s ) Pr ( T > s ) = Pr ( T > s + t ) Pr ( T > s ) = e − λ ( s + t ) e − λ s = e − λ t = Pr ( T > t ) . {\displaystyle {\begin{aligned}\Pr \left(T>s+t|T>s\right)&={\frac {\Pr \left(T>s+t\cap T>s\right)}{\Pr \left(T>s\right)}}\\&={\frac {\Pr \left(T>s+t\right)}{\Pr \left(T>s\right)}}\\&={\frac {e^{-\lambda (s+t)}}{e^{-\lambda s}}}\\&=e^{-\lambda t}\\&=\Pr(T>t).\end{aligned}}}
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 ) = E λ 0 ( log  p λ 0 ( x ) p λ ( x ) ) = E λ 0 ( log  λ 0 e − λ 0 x λ e − λ x ) = log  ( λ 0 ) − log  ( λ ) − ( λ 0 − λ ) E λ 0 ( x ) = log  ( λ 0 ) − log  ( λ ) + λ λ 0 − 1. {\displaystyle {\begin{aligned}\Delta (\lambda _{0}||\lambda )&=\mathbb {E} _{\lambda _{0}}\left(\log {\frac {p_{\lambda _{0}}(x)}{p_{\lambda }(x)}}\right)\\&=\mathbb {E} _{\lambda _{0}}\left(\log {\frac {\lambda _{0}e^{-\lambda _{0}x}}{\lambda e^{-\lambda x}}}\right)\\&=\log(\lambda _{0})-\log(\lambda )-(\lambda _{0}-\lambda )E_{\lambda _{0}}(x)\\&=\log(\lambda _{0})-\log(\lambda )+{\frac {\lambda }{\lambda _{0}}}-1.\end{aligned}}}
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]
Fisher Information[edit]
The Fisher information, denoted I ( λ ) {\displaystyle {\mathcal {I}}(\lambda )} , for an estimator of the rate parameter λ {\displaystyle \lambda } is given as:
I ( λ ) = E  [ ( ∂ ∂ λ log  f ( x ; λ ) ) 2 | λ ] = ∫ ( ∂ ∂ λ log  f ( x ; λ ) ) 2 f ( x ; λ ) d x {\displaystyle {\mathcal {I}}(\lambda )=\operatorname {E} \left[\left.\left({\frac {\partial }{\partial \lambda }}\log f(x;\lambda )\right)^{2}\right|\lambda \right]=\int \left({\frac {\partial }{\partial \lambda }}\log f(x;\lambda )\right)^{2}f(x;\lambda )\,dx}
Plugging in the distribution and solving gives:
I ( λ ) = ∫ 0 ∞ ( ∂ ∂ λ log  λ e − λ x ) 2 λ e − λ x d x = ∫ 0 ∞ ( 1 λ − x ) 2 λ e − λ x d x = λ − 2 {\displaystyle {\mathcal {I}}(\lambda )=\int _{0}^{\infty }\left({\frac {\partial }{\partial \lambda }}\log \lambda e^{-\lambda x}\right)^{2}\lambda e^{-\lambda x}\,dx=\int _{0}^{\infty }\left({\frac {1}{\lambda }}-x\right)^{2}\lambda e^{-\lambda x}\,dx=\lambda ^{-2}} .
This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter λ {\displaystyle \lambda } .
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 α 2 , 2 n 2 < 1 λ < 2 n λ ^ χ 1 − α 2 , 2 n 2 {\displaystyle {\frac {2n}{{\widehat {\lambda }}\chi _{{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n}{{\widehat {\lambda }}\chi _{1-{\frac {\alpha }{2}},2n}^{2}}}}
which is also equal to:
2 n x ¯ χ α 2 , 2 n 2 < 1 λ < 2 n x ¯ χ 1 − α 2 , 2 n 2 {\displaystyle {\frac {2n{\overline {x}}}{\chi _{{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n{\overline {x}}}{\chi _{1-{\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). 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.
Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using CumFreq[11]
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: \frac{\lambda}{\lambda-t}, \text{ for } t < \lambda

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values:
 f(x;\lambda) = \begin{cases}
\lambda e^{-\lambda x} & x \ge 0, \\
0 & x < 0.
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values: {\displaystyle f(x;\lambda )=\mathrm {\lambda } e^{-\lambda x}H(x)}

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values: f(x;\lambda) = \mathrm \lambda e^{-\lambda x} H(x)

values: Edit section: Cumulative distribution function

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values: {\displaystyle F(x;\lambda )=\mathrm {(} 1-e^{-\lambda x})H(x)}

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values: F(x;\lambda) = \mathrm (1-e^{-\lambda x}) H(x)

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values:
f(x;\beta) = \begin{cases}
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First moment

values: {\displaystyle \operatorname {E} [X]={\frac {1}{\lambda }}=\beta }

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values: \operatorname{E}[X] = \frac{1}{\lambda} = \beta

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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: F^{-1}(p;\lambda) = \frac{-\ln(1-p)}{\lambda},\qquad 0 \le p < 1

values: Edit section: Kullback–Leibler divergence

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values: Edit section: Maximum entropy distribution

values: Maximum entropy probability distribution

values: Edit section: Distribution of the minimum of exponential random variables

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Independent random variables

values: {\displaystyle \min \left\{X_{1},\dotsc ,X_{n}\right\}}

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values: {\displaystyle \lambda =\lambda _{1}+\dotsb +\lambda _{n}}

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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).
\end{align}

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values: Edit section: Joint moments of i.i.d. exponential order statistics

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values: Independent and identically distributed random variables

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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),

values: {\displaystyle {\overline {x}}={1 \over n}\sum _{i=1}^{n}x_{i}}

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values: \overline{x}={1 \over n}\sum_{i=1}^n x_i

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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: {\displaystyle {\mathcal {I}}(\lambda )}

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Chi squared distribution

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values: \lambda_{low}=\widehat{\lambda}  \left (1-\frac{1.96}{\sqrt{n}} \right )

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values: \lambda_{upp}=\widehat{\lambda}  \left (1+\frac{1.96}{\sqrt{n}} \right )

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values: \mathrm{Gamma}(\lambda; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda\beta).

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Posterior distribution

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values:
 \begin{align}
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{align}

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values: p(\lambda) = \mathrm{Gamma}(\lambda; \alpha + n, \beta + n \overline{x}).

values: {\displaystyle {\frac {\alpha +n}{\beta +n{\overline {x}}}}.}

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values: \frac{\alpha + n}{\beta + n \overline{x}}.

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Inverse transform sampling method

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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) \\
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Bosch (TV series)
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Bosch
Genre
Detective fiction
Drama
Created by
Michael Connelly
Developed by
Eric Overmyer
Starring
Titus Welliver
Jamie Hector
Amy Aquino
Lance Reddick
Annie Wersching
Sarah Clarke
Jason Gedrick
Madison Lintz
Jeri Ryan
Brent Sexton
Opening theme
"Can't Let Go" by Caught A Ghost
Country of origin
United States
Original language(s)
English
No. of seasons
3
No. of episodes
30 (list of episodes)
Production
Cinematography
Eric Alan Edwards
Running time
50 minutes
Release
Original network
Amazon Video
Original release
February 6, 2014 (2014-02-06) – present
External links
Website
www.amazon.com/Pilot-HD/dp/B00I3MPDP4
Bosch is an American police procedural web television series produced by Amazon Studios and Fabrik Entertainment. It stars Titus Welliver as Los Angeles Police detective Harry Bosch. The show was developed for Amazon by Eric Overmyer[1] and the first season takes its inspiration from three of Michael Connelly’s novels: City of Bones, Echo Park, and The Concrete Blonde.
It is one of two drama pilots that Amazon streamed online in early 2014. Viewers were allowed to offer their opinions about the pilot before the studio decided whether to place a series order.[2]
On March 12, 2014, Amazon.com ordered a full season to appear on Amazon Prime, and the season premiered on February 13, 2015.[3]
On March 18, 2015, Bosch was renewed for a second season,[4] which takes inspiration from Connelly's novels Trunk Music, The Drop, and The Last Coyote.[5] The second season premiered on March 11, 2016.
A third season, which adapts Connelly's novel The Black Echo and elements of A Darkness More Than Night, premiered on April 21, 2017. On April 24, 2017, Connelly confirmed on his official Facebook page that a fourth season is currently being written.[6] According to Connelly, the plot will be largely based on Angels Flight with elements from Nine Dragons.[citation needed]
Contents
[hide]
1 Plot
1.1 Season 1
2 Production
3 Cast
3.1 Main cast
3.1.1 Season 1
3.1.2 Season 2
3.2 Recurring cast
3.2.1 Season 1
3.2.2 Season 2
3.2.3 Season 3
4 Episodes
5 Reception
6 References
7 External links
Plot[edit]
Season 1[edit]
As the pilot opens, Bosch is tailing a suspect. Eventually cornering him in an alley, Bosch shoots the suspect when he reaches in his pocket. The incident is shown later in the episode in two separate flashbacks. When seen from Bosch's point of view it appears that there is something in the suspect's hand that falls in a puddle. When the incident is recounted by the plaintiff's lawyer during a wrongful death suit, there is clearly nothing in the suspect's hand and Bosch is shown planting a gun. Whatever really happened, he is cleared by the department. The show fast-forwards to two years later where Bosch is being sued by the family of the suspect in a wrongful death civil suit.
Feeling that he has to do something as a police officer, he agrees to trade with two other detectives to take the weekend shift, where he is called out on a case which turns out to be a suicide, and a second case where a doctor reports his dog found a human bone in the woods.
The bone leads to more bones and the coroner determines the skeleton is that of a small boy who was horribly abused and beaten, then buried in the woods. The boy has been dead since at least 1989, and could have been anything from 10 to 12 when he died, but was so horribly treated that it is not certain exactly how old he was. The details of the boy's mistreatment – more than 40 broken bones, some having healed while others were relatively recent – and his death are so grisly that Bosch has to step away and go into the restroom to splash water on his face and sit down on a commode for a moment to regain his composure.
Production[edit]
Amazon Studios announced on October 31, 2013 that it had given the green light to Bosch for production. The hour-long pilot starred Titus Welliver as Harry Bosch, with Annie Wersching, Amy Price-Francis, and Jamie Hector co-starring. Henrik Bastin of Fabrik Entertainment was the producer and Jim McKay directed.[7]
According to Connelly, "a fair amount of changes" were made "to the world of Harry Bosch" "in making the shift from page to screen." In the series, Harry "is 47 years old and a veteran of the first Gulf War in 1991," when he was a member of a Special Forces team clearing tunnels, but "he has now been a police officer for twenty years, with a one-year exception when he re-upped with the Army after 9/11, as many LAPD officers did. He came back to the force after serving in Afghanistan and again encountering tunnel warfare."[8]
On November 4, 2013, the 13-day shoot began in Los Angeles, while Connelly kept a daily set journal.[9]
The pilot premiered on Amazon Prime in February 2014 to allow customers to vote to decide if more episodes should be made.[10] In March 2014, Amazon announced that they had commissioned Bosch for a full series.[11]
All ten episodes of the first season of Bosch were released for viewing on Amazon Video on February 13, 2015.[12] Portions of the first episode were changed from the pilot, including the addition of Mimi Rogers to the cast to replace Amy Price-Francis as plaintiff's attorney Honey Chandler and the addition of a scene in which Bosch testifies in court and is questioned about his background by Chandler.
On March 18, 2015, Bosch was renewed for a second season.[4] On July 16, the series was nominated for the Outstanding Main Title Design award at the 67th Primetime Creative Arts Emmy Awards, along with Manhattan, American Horror Story: Freak Show, Daredevil, Halt and Catch Fire, and Olive Kitteridge; the award was won by Manhattan.[13]
On April 1, 2016, Bosch was renewed for a third season.[14][15] On October 16, 2016, Bosch was renewed for a fourth season.[16]
Cast[edit]
Main cast[edit]
Titus Welliver as Harry Bosch
Titus Welliver as Los Angeles Police Department Detective Hieronymus 'Harry' Bosch, a homicide detective in the Hollywood Division. Something of a renegade, Harry is an astute detective with a fundamental respect for rules and policy. Harry lives in a house on stilts in the Hollywood Hills, purchased with money he earned as a technical advisor on a film. (Quinn Welliver as adolescent Harry.)[citation needed]
Jamie Hector as Det. Jerry Edgar, Harry's partner
Amy Aquino as Lieutenant Grace Billets, Harry's immediate superior and his friend
Lance Reddick as Deputy Chief Irvin Irving, West Bureau Commanding Officer
Madison Lintz as Maddie Bosch, Harry's teenaged daughter. She has a close, loving relationship with her father (recurring season 1; main season 2-)
Sarah Clarke as Eleanor Wish, Harry's ex-wife, with whom he still has a cordial, if tense, relationship (season 1–2; guest season 3)
Season 1[edit]
Annie Wersching as Officer Julia Brasher, a rookie cop assigned to the Hollywood Division. She becomes involved with Harry, but comes into conflict with him when he realizes she makes up the rules as she goes along.
Jason Gedrick as Raynard Waits, a serial killer and the suspect in the death of a boy whose bones are found in Laurel Canyon
Season 2[edit]
Jeri Ryan as Veronica Allen, a manipulative former porn star married to an Armenian porn producer who is murdered.
Brent Sexton as Carl Nash, a corrupt LAPD Homicide Detective, now retired, who oversees a team of villainous police officers while working as a security guard at the estate where the Allens live.
Recurring cast[edit]
Scott Klace as Sgt. John Mankiewicz, a Sergeant and Assistant Watch Commander at the Hollywood Division station
Troy Evans as Det. Johnson (Barrel), a senior homicide detective at Hollywood Division
Gregory Scott Cummins as Det. Moore (Crate), Barrel's longtime friend and partner
David Marciano as Det. Brad Conniff, Robbery Homicide Division (season 2-)
DaJuan Johnson as Off., later Det. Rondell Pierce
Robbie Jones as Off. George Irving, Deputy Chief Irving's son, a rookie cop later assigned to undercover narcotics (season 1–2)
Steven Culp as Richard 'Rick' O'Shea, the politically ambitious Los Angeles County District Attorney
Mimi Rogers as Honey 'Money' Chandler, a civil rights attorney
Matthew Lillard as Luke 'Lucky' Rykov, an FBI agent with whom Harry works on two cases
Yancey Arias as Los Angeles Mayor Hector Ramos (season 2-)
Season 1[edit]
Scott Wilson as Dr. Paul Guyot, a retired doctor whose dog found bones of a missing boy.
Alan Rosenberg as Dr. William Golliher, a forensic anthropologist who assisted Bosch with the identification of the bones
Mark Derwin as Captain Harvey Pounds
Abraham Benrubi as Rodney Belk, a lawyer who represented Bosch in his trial.
Paul Vincent O'Connor as Judge Alan M. Keyes, a judge who presided at Bosch's trial.
Adam O'Byrne as Nate Tyler, an aggressive LA Times reporter
Veronica Cartwright as Janet Saxon, Reynard Waits' mother
Deji LaRay as Off. Julius Edgewood
Jason Sims-Prewitt as Off. Victor Rhodes
Rose Rollins as Det. Kizmin Rider
Shawn Hatosy as Johnny Stokes
Season 2[edit]
Erika Alexander as Connie Irving, Deputy Chief Irving's wife and Off. Irving's mother
James Ransone as Off. Eddie Arceneaux, Off. Irving's partner, involved in the ring of dirty cops led by Carl Nash
Leisha Hailey as Off. Maureen 'Mo' O'Grady, also a cop in Nash's ring
Emilia Zoryan as Layla, the stagename of a dancer at Dolly's in Las Vegas, Nevada and mysterious girlfriend to an Armenian porn producer who is murdered.
Season 3[edit]
Paul Calderón as Detective Santiago Robertson, a seasoned detective investigating the murder of a vet who has a history with Harry
Paola Turbay as DDA Anita Benitez, who is trying the Holland case, and has a brief relationship with Harry
Bridger Zadina as Sharkey, a young street boy who is key to one of Harry's cases
Eric Ladin as Scott Anderson, a reporter for the Los Angeles Times following the Gunn murder case
John Ales as Andrew Holland, a movie writer/director accused of murder
Spencer Garrett as Fowkkes, Andrew Holland's high-powered attorney
Arnold Vosloo as Rudy Tafero, an investigator working for the defense on the Holland case
Jared Ward as Jesse Tafero, Rudy's younger brother and helper
Beth Broderick as Judge Sharon Houghton
Jeffrey Pierce as Trevor Dobbs, leader of a team of former military men transiting illicit cargo through the Port of Los Angeles
Max Arciniega as Xavi Moreno, a sniper who is part of Dobbs' team
Christopher Backus as Woody Woodrow, part of Dobbs' team
Brooke Smith as Captain Ellen Lewis, Hollywood Station
Monti Sharp as the mysterious man on a bicycle involved in the Korea Town Killer (KTK) murders
Episodes[edit]
Main article: List of Bosch episodes
Season
Episodes
Originally released
1 10 February 13, 2015 (2015-02-13)
2 10 March 11, 2016 (2016-03-11)
3 10 April 21, 2017 (2017-04-21)
Reception[edit]
Reviews of Bosch have been generally positive. Metacritic gives the series a score average of 71 (out of 100) based on 16 critics.[17]
Cory Barker of TV.com wrote that the series is "rock-solid and generally enjoyable without ever making much of an attempt to push boundaries," and praised Amazon Studios for "producing a show based on a book that somehow reproduces the experience of reading."[18]
Neil Genzlinger of The New York Times wrote that, despite an "impossibly long" "list of brooding, taciturn small-screen police detectives," Bosch "proves gripping" because its "plotting and pacing...draw you in...like a good page-turner."[19]
Noel Murray of the A.V. Club remarked that "the best thing about Bosch is how well it captures Connelly’s Los Angeles," while noting that "the series’ biggest stumbling block is that it’s stubbornly slow-paced" despite the fact that "the slow-drip approach makes sense."[20]
Brian Lowry of Variety wrote that "the series has the texture and tone of an old-fashioned detective yarn," but said that "the transition from page to screen ... proves too talky in places and clunky in others". He remarked that the show boasted "good casting and a strong sense of L.A. noir", but opines that the series "feels undercooked".[21]
Hank Stuever of The Washington Post called Welliver's performance "nicely built out of smirks and smolders," and wrote that the show's "unvarnished, unglamorous L.A. is a far more intriguing, far more complex setting for a story."[22]
Brian Moylan of The Guardian praised the "noir" feeling of the show and considered it a step above NCIS, but did not like the similarities to too many other cop shows, calling the series "samey".[3]
References[edit]
Jump up ^ Patten, Dominic. "Amazon Renews 'Bosch' for Second Season". Deadline.com. Penske Business Media, LLC. Retrieved March 22, 2015.
Jump up ^ Porter, Rick (October 31, 2013). "Amazon Orders Michael Connelly Pilot 'Bosch,' Details 'X-Files' Creator's 'The After'". Zap2It. Archived from the original on November 3, 2013. Retrieved December 4, 2013.
^ Jump up to: a b Moylan, Brian. "Amazon's Bosch: Paint-by-Numbers Cop Show that's way past its prime". The Guardian.
^ Jump up to: a b Spangler, Todd (March 18, 2015). "Amazon Renews 'Bosch' Cop Drama for Season 2". Variety. Retrieved July 26, 2017.
Jump up ^ Braxton, Greg (March 18, 2015). "'Bosch' series gets second season shortly after Amazon Prime premiere". Los Angeles Times. Retrieved March 18, 2015.
Jump up ^ "season 4 confirmation on Michael Connelly's official Facebook page". Facebook. April 24, 2017. Retrieved April 24, 2017.
Jump up ^ "Amazon - Press Room - Press Release". Phx.corporate-ir.net. 2013-10-31. Retrieved 2017-07-26.
Jump up ^ "Bosch Tv". MichaelConnelly.com. Retrieved 2017-07-26.
Jump up ^ Michael Connelly's Set Journal Archived 2013-11-12 at the Wayback Machine.
Jump up ^ "Amazon Launches Second Pilot Season". BBC News. February 7, 2014. Retrieved March 23, 2014.
Jump up ^ "Amazon Studios "Orders Four Original Series£". BBC News. March 12, 2014. Retrieved March 23, 2014.
Jump up ^ "Countdown to 'Bosch': Amazon's First Drama to... – Hollywonk". Hollywonk. Archived from the original on 2015-01-16.
Jump up ^ "67th Emmy Awards Nominees and Winners". Television Academy. July 16, 2015. Retrieved November 23, 2015.
Jump up ^ Connelly, Michael (April 3, 2016). "Bosch Season 3 Is A Go!". MichaelConnelly.com. Retrieved April 3, 2016.
Jump up ^ Patten, Dominic (2016-04-01). "'Bosch' Renewed For Season 3 By Amazon". Deadline.com. Retrieved 2017-07-26.
Jump up ^ "'Bosch' Renewed For Season 4". IsMyShowCancelled.com. 2016-10-17. Retrieved 2017-07-26.
Jump up ^ "Critic Reviews for Bosch Season 1". Metacritic. Retrieved 2017-07-26.
Jump up ^ Cory Barker, "Amazon's Newest Drama Is Like a Decent Book You Can't Put Down," TV.com
Jump up ^ Neil Genzlinger, "‘Bosch,’ Amazon Prime’s New Crime Series," The New York Times
Jump up ^ Noel Murray, "Amazon’s Bosch brings a beloved pulp hero to the screen," The A.V. Club
Jump up ^ Brian Lowry, "TV Review: Amazon’s ‘Bosch’," Variety
Jump up ^ Hank Stuever, "‘Bosch’: Yet another doleful detective, but this one might have a case," The Washington Post
External links[edit]
Bosch on IMDb
[show]
v
t
e
Novels by Michael Connelly
Harry Bosch series
The Black Echo (1992)
The Black Ice (1993)
The Concrete Blonde (1994)
The Last Coyote (1995)
Trunk Music (1997)
Angels Flight (1999)
A Darkness More Than Night (2001)
City of Bones (2002)
Lost Light (2003)
The Narrows (2004)
The Closers (2005)
Echo Park (2006)
The Overlook (2007)
9 Dragons (2009)
The Drop (2011)
The Black Box (2012)
The Burning Room (2014)
The Crossing (2015)
The Wrong Side of Goodbye (2016)
Mickey Haller series
The Lincoln Lawyer (2005)
The Brass Verdict (2008)
The Reversal (2010)
The Fifth Witness (2011)
The Gods of Guilt (2013)
Jack McEvoy series
The Poet (1996)
The Scarecrow (2009)
Other novels
Blood Work (1998)
Void Moon (2000)
Chasing the Dime (2002)
Adaptations
Blood Work (2002)
The Lincoln Lawyer (2011)
Bosch (2015)
[show]
v
t
e
Amazon Video original programming
Original series
Alpha House (2013–14)
Betas (2013–14)
Bosch (2014–present)
Mozart in the Jungle (2014–present)
Transparent (2014–present)
Hand of God (2014–17)
The Man in the High Castle (2015–present)
One Mississippi (2015–present)
Patriot (2015–present)
Red Oaks (2015–present)
Sneaky Pete (2015–present)
Z: The Beginning of Everything (2015–17)
Mad Dogs (2015–16)
Crisis in Six Scenes (2016)
Good Girls Revolt (2016)
The New Yorker Presents (2016)
Kamen Rider Amazons
The Collection (2016–present)
Crayon Shin-chan Gaiden (2016–present)
Fleabag (2016–present)
Goliath (2016–present)
The Grand Tour (2016–present)
I Love Dick (2016–present)
Jean-Claude Van Johnson (2016–present)
The Last Tycoon (2016–17)
The Tick (2016–present)
American Playboy: The Hugh Hefner Story (2017)
The Idolmaster KR (2017)
Inside Edge (2017–present)
The Marvelous Mrs. Maisel (2017–present)
Comrade Detective (2017–present)
Continuations
Ripper Street (series 3–5; 2014–16)
Children's
programming
Creative Galaxy (2013–present)
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This page was last edited on 12 November 2017, at 20:06.
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