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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 the 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
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
2.8 Sum of two independent exponential random variables
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 )=\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\mid 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\mid T>s\right)&={\frac {\Pr \left(T>s+t\cap T>s\right)}{\Pr \left(T>s\right)}}\\[4pt]&={\frac {\Pr \left(T>s+t\right)}{\Pr \left(T>s\right)}}\\[4pt]&={\frac {e^{-\lambda (s+t)}}{e^{-\lambda s}}}\\[4pt]&=e^{-\lambda t}\\[4pt]&=\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 after 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\mid 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 ) ] = ∫ 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}\operatorname {E} [X_{(i)}X_{(j)}]&=\int _{0}^{\infty }\operatorname {E} [X_{(i)}X_{(j)}\mid X_{(i)}=x]f_{X_{(i)}}(x)\,dx\\&=\int _{x=0}^{\infty }x\operatorname {E} [X_{(j)}\mid 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 }}\operatorname {E} [X_{(i)}]+\operatorname {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 \operatorname {E} [X_{(j)}\mid X_{(j)}\geq x]} with E  [ X ( j ) ] + x {\displaystyle \operatorname {E} [X_{(j)}]+x} .
Sum of two independent exponential random variables[edit]
The sum of two independent variables corresponds to the convolution of probability distributions. If X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are independent exponential random variables of independent observations, with respective rate parameters λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} , then the probability density of Z = X 1 + X 2 {\displaystyle Z=X_{1}+X_{2}} is given by,
f Z ( z ) = ∫ 0 z f X 1 ( x 1 ) f X 2 ( z − x 1 ) d x 1 = λ 1 λ 2 exp  [ − λ 2 z ] ∫ 0 z exp  [ ( λ 2 − λ 1 ) x 1 ] d x 1 = λ 1 λ 2 λ 2 − λ 1 ( exp  [ − λ 1 z ] − exp  [ − λ 2 z ] ) {\displaystyle {\begin{aligned}f_{Z}(z)=&\int _{0}^{z}f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})\,dx_{1}\\=&\lambda _{1}\lambda _{2}\exp[-\lambda _{2}z]\int _{0}^{z}\exp[(\lambda _{2}-\lambda _{1})x_{1}]\,dx_{1}\\=&{\frac {\lambda _{1}\lambda _{2}}{\lambda _{2}-\lambda _{1}}}\left(\exp[-\lambda _{1}z]-\exp[-\lambda _{2}z]\right)\end{aligned}}}
The mean and variance of Z = X 1 + X 2 {\displaystyle Z=X_{1}+X_{2}} are found to be,
E  [ Z ] = ∫ z d f Z ( z ) = λ 1 − 1 + λ 2 − 1 var  ( Z ) = E  [ Z 2 ] − E  [ Z ] 2 = ∫ z 2 d f Z ( z ) − ( λ 1 − 1 + λ 2 − 1 ) 2 = λ 1 − 2 + λ 2 − 2 {\displaystyle {\begin{aligned}\operatorname {E} [Z]=&\int z\,df_{Z}(z)\\[4pt]=&\lambda _{1}^{-1}+\lambda _{2}^{-1}\\[4pt]\operatorname {var} (Z)=&\operatorname {E} [Z^{2}]-\operatorname {E} [Z]^{2}\\[4pt]=&\int z^{2}\,df_{Z}(z)-(\lambda _{1}^{-1}+\lambda _{2}^{-1})^{2}\\[4pt]=&\lambda _{1}^{-2}+\lambda _{2}^{-2}\end{aligned}}}
The cumulative density function of the sum of two independent exponential random variables then follows as,
F Z ( z ) = Pr ( Z ≤ z ) = ∫ 0 z f Z ( t ) d t = 1 + λ 1 λ 2 − λ 1 exp  [ − λ 2 z ] − λ 2 λ 2 − λ 1 exp  [ − λ 1 z ] {\displaystyle {\begin{aligned}F_{Z}(z)=\Pr(Z\leq z)=&\int _{0}^{z}f_{Z}(t)\,dt\\[4pt]=&1+{\frac {\lambda _{1}}{\lambda _{2}-\lambda _{1}}}\exp[-\lambda _{2}z]-{\frac {\lambda _{2}}{\lambda _{2}-\lambda _{1}}}\exp[-\lambda _{1}z]\end{aligned}}}
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 lower = λ ^ ( 1 − 1.96 n ) {\displaystyle \lambda _{\text{lower}}={\widehat {\lambda }}\left(1-{\frac {1.96}{\sqrt {n}}}\right)}
λ upper = λ ^ ( 1 + 1.96 n ) {\displaystyle \lambda _{\text{upper}}={\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:
Gamma  ( λ ; α , β ) = β α Γ ( α ) λ α − 1 exp  ( − λ β ) . {\displaystyle \operatorname {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 \operatorname {Exp} \left(\lambda \right)={\frac {1}{2\lambda }}\operatorname {Exp} \left({\frac {1}{2}}\right)\sim {\frac {1}{2\lambda }}\chi _{2}^{2}\Rightarrow \sum \limits _{i=1}^{n}\operatorname {Exp} (\lambda )\sim {\frac {1}{2\lambda }}\chi _{2n}^{2}}
If Xi ~ Exp(λi) then min{X1, ..., Xn} ~ Exp(λ1+ ... +λn).
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 Beta  ( 1 , n ) = Exp  ( 1 ) . {\displaystyle \lim _{n\to \infty }n\operatorname {Beta} (1,n)=\operatorname {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 ∼ Rayleigh  ( 1 / 2 λ ) . {\displaystyle {\sqrt {X}}\sim \operatorname {Rayleigh} \left(1/{\sqrt {2\lambda }}\right).} (Rayleigh distribution)
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
o p e r a t o r n a m e E λ 0 [ Δ ( λ 0 ∣∣ p M L ) ] = ψ ( n ) + 1 n − 1 − log  ( n ) o p e r a t o r n a m e E λ 0 [ Δ ( λ 0 ∣∣ p C N M L ) ] = ψ ( n ) + 1 n − log  ( n ) {\displaystyle {\begin{aligned}operatorname{E}_{\lambda _{0}}\left[\Delta (\lambda _{0}\mid \mid p_{\rm {ML}})\right]&=\psi (n)+{\frac {1}{n-1}}-\log(n)\\operatorname{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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 f(x;\lambda) = \begin{cases}
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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: {\displaystyle \operatorname {E} [X_{(j)}\mid X_{(j)}\geq x]}

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

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 \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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values: {\displaystyle F^{-1}(p)={\frac {-\ln(1-p)}{\lambda }}.}

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values: {\displaystyle T={\frac {-\ln(U)}{\lambda }}.}

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Benktander Weibull distribution

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values: \mu-\beta\log \left(\tfrac{e^{-X}}{1-e^{-X}}\right) \sim \mathrm{Logistic}(\mu,\beta)

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values: {\displaystyle {\tfrac {e^{-X}}{k}}\sim \mathrm {PowerLaw} (k,\lambda )}

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values: \tfrac{e^{-X}}{k} \sim \mathrm{PowerLaw}(k, \lambda)

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values: {\displaystyle X\sim \mathrm {Weibull} ({\tfrac {1}{\lambda }},1)}

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values: \lim_{n \to \infty}n \min \left (X_1, \ldots, X_n \right ) \sim \textrm{Exp}(1)

values: {\displaystyle Y\sim \mathrm {Geometric} ({\tfrac {1}{1+\lambda }})}

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values: {\displaystyle Y\sim \Gamma (\alpha ,{\tfrac {\beta }{\alpha }})}

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values: {\displaystyle {\sqrt {XY}}\sim \mathrm {K} (\alpha ,\beta )}

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Hoyt distribution

values: {\displaystyle {\frac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}

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values: {\displaystyle Y\sim \Gamma (n,{\tfrac {1}{\lambda }})}

values: vertical-align: -1.338ex; width:12.357ex; height:3.676ex;

values: {\displaystyle {\tfrac {X}{Y}}+1\sim \mathrm {Pareto} (1,n)}

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Skew-logistic distribution

values: {\displaystyle Z={\frac {\lambda _{X}X}{\lambda _{Y}Y}}}

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values: {\displaystyle {\frac {\lambda _{X}}{\lambda _{Y}}}}

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Hyper-exponential distribution

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ExGaussian distribution

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Poisson process

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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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text:
Doll on a Music Box
From Wikipedia, the free encyclopedia
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This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) (Learn how and when to remove this template message)
"Doll on a Music Box" is a song originally from the 1968 musical film, Chitty Chitty Bang Bang. It was subsequently performed in the 2002/2005 stage musical Chitty Chitty Bang Bang as well. It is both a musical and lyrical counterpoint to the more free flowing, legato song, "Truly Scrumptious". In the song, Truly is disguised as a wind up music box doll, metaphorically and actually on a pedestal. In the song, Truly sings about herself and her rigid nature, all behind the mask of the "doll" she is portraying. In the motion picture the part of Truly was played by actress Sally Ann Howes. In the stage musical version, the part was re-created by 19-year-old London actress, Emma Williams. In 2005, the Broadway "Truly" was portrayed by actress Erin Dilly, who was nominated for a Tony Award that year for the role.
Contents
[hide]
1 Visual metaphor
2 Songwriters
3 Trivia
4 Wardrobe
5 References
Visual metaphor[edit]
This sequence is a part of Caractacus's story, which he is telling to Truly and his children. In the song, Caractacus plays a marionette, too flexible for his own good; whereas, Truly plays a precious, too-cutesy doll that is too rigid for her own good. Twice during the counterpoint, the clown approaches the doll, only to be slapped in the face. This mirrors the earlier story where Caractacus's first two interactions with Truly are mistimed as well. The clown keeps trying, when suddenly he captures his own reflection and realizes how foolish he appears. It is only when Truly momentarily breaks character and shoves him slightly into the mirror to get his attention that he returns to his character. The clown properly courts the doll and is about to kiss the doll's hand, when she retracts her hand from him just in time. Rigidity wins out. Only at the end of the movie do Caractacus and Truly truly see eye to eye.
Songwriters[edit]
The song was written by Robert B. Sherman & Richard M. Sherman.
Trivia[edit]
In 1992, "Doll on a Music Box" was re-enacted as part of a surrealistic sequence of the 1992 film, Arizona Dream starring Johnny Depp, Jerry Lewis, Faye Dunaway and Lili Taylor.
The music video for the Barenaked Ladies 1998 single, "One Week" had a short reference to "Doll on a Music Box".
Wardrobe[edit]
Truly Scrumptious sports a dirndl in this sequence consisting of a forest green satin bodice (trimmed in black velvet and accented with stitched crisscrossed red ribbon straps down the front and gold buttons), white blouse (the sleeves are double puffed done in delicate white silk satin; there are three yellow ribbons stitched to the sleeves for added detail; the collar is also trimmed in lace), full, printed (in the shape of gold diamonds) cotton red skirt (a tea length petticoat) and white, cotton apron (with a German inspired floral motif). Truly also wore white tights and black pumps (with a silver buckle near the toe area) with a Tyrolean hairstyle incorporating both pigtails and braids.
References[edit]
[hide]
v
t
e
Ian Fleming's Chitty Chitty Bang Bang
Characters
Caractacus Pott
Truly Scrumptious
Child Catcher
Objects and locations
Vulgaria
Chitty Chitty Bang Bang (car)
Chitty Bang Bang (airship)
Media
Chitty-Chitty-Bang-Bang: The Magical Car (1964 book)
Chitty Chitty Bang Bang (film)
"Chitty Chitty Bang Bang" (song)
Chitty Chitty Bang Bang (musical)
Other
Chitty Bang Bang (original car)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Doll_on_a_Music_Box&oldid=802844336"
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text:
The whole nine yards
From Wikipedia, the free encyclopedia
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This article is about the phrase. For the film, see The Whole Nine Yards (film).
A 1921 headline from The Spartanburg Herald-Journal in South Carolina reads "The Whole Six Yards of It."
The whole nine yards or the full nine yards is a colloquial American phrase meaning "everything, the whole lot" or, when used as an adjective, "all the way", as in, "The Army came out and gave us the whole nine yards on how they use space systems."[1] Its origin is unknown and has been described by Yale University librarian Fred R. Shapiro as "the most prominent etymological riddle of our time".[2]
The Oxford English Dictionary finds the earliest published non-idiomatic use in an 1855 Indiana newspaper article. The earliest known idiomatic use of the phrase is from 1907 in southern Indiana. The phrase is related to the expression the whole six yards, used around the same time in Kentucky and South Carolina. Both phrases are variations on the whole ball of wax, first recorded in the 1880s.[3] They are part of a family of expressions in which an odd-sounding item, such as enchilada, shooting match, shebang or hog, is substituted for ball of wax.[3] The choice of the number nine may be related to the expression "To the nines" (to perfection).[nb 1]
Use of the phrase became widespread in the 1980s and 1990s. Much of the interest in the phrase's etymology can be attributed to New York Times language columnist William Safire, who wrote extensively on this question.
Contents
[hide]
1 History of the phrase
2 Research
3 Origin
4 See also
5 Notes
6 References
History of the phrase[edit]
The Oxford English Dictionary places the earliest published non-idiomatic use of the phrase in the New Albany (Indiana) Daily Ledger, January 30, 1855 in an article called "The Judge's Big Shirt." “What a silly, stupid woman! I told her to get just enough to make three shirts; instead of making three, she has put the whole nine yards into one shirt!” The first known use of the phrase as an idiom appears in The Mitchell Commercial, a newspaper in the small town of Mitchell, Indiana, in its May 2, 1907 edition:
This afternoon at 2:30 will be called one of the baseball games that will be worth going a long way to see. The regular nine is going to play the business men as many innings as they can stand, but we can not promise the full nine yards.[4]
The idiom was used three more times in the Mitchell Commercial over the next seven years, in the forms give him the whole nine yards (i.e., tell someone a big story),[5] take the whole nine yards (i.e., take everything),[6] and settled the whole nine yards (i.e., resolved everything).[7]
In other uses from this time period, the phrase was given as the whole six yards. In 1912, a local newspaper in Kentucky asked readers to, "Just wait boys until the fix gets to a fever heat and they will tell the whole six yards."[8] The same newspaper repeated the phrase soon afterward in another issue, stating "As we have been gone for a few days and failed to get all the news for this issue we will give you the whole six yards in our next." [9] The six-yard form of the phrase also appears in a 1921 headline in a local South Carolina paper.[10]
The phrase is not known to have been used in writing thereafter until a 1956 issue of Kentucky Happy Hunting Ground, where it appears in an article on fishing.[11][12] After describing the contests and prizes, the author writes, "So that's the whole nine-yards."[13][14] It appeared in an article on hunting the following year, this time unhyphenated.[12]
The phrase at this point was still rare. There is strong circumstantial evidence it was not in general use in 1961, as Ralph Boston set a world record for the long jump that year at 27 feet, or nine yards, but no news report has been found that made any reference to the term, suggesting that journalists were unaware of it or did not regard it as common enough to use as a pun.[15]
In a short story published in 1962, the phrase is attributed to "a brush salesman".[16] A letter published in an auto magazine later that year describes a certain new car as containing "all nine yards of goodies".[17] In 1964, several newspapers published a syndicated story which explained that "Give 'em the whole nine yards" was NASA talk for an item-by-item report.[18] This early usage can be read as suggesting length, but can also be read as suggesting detailed completeness.[19]
Two 1965 newspaper articles quote U.S. military personnel serving in Vietnam using the phrase.[20] The phrase was explained as something "teenagers say" in a military-oriented magazine in 1965.[21] Citations from 1966 show the phrase was used by a former U.S. Army airman,[22] and also in a publication for military test pilots.[23] It is also recorded in two contemporary novels concerning the U.S. Air Force in Vietnam, Carl Krueger's Wings of a Tiger (1966),[24] and Elaine Shepard's The Doom Pussy (1967).[25]
Research[edit]
William Safire, a language columnist at the New York Times, asked listeners for information regarding the origin of the phrase on Larry King's radio show in 1982.[11] Safire ended up writing nine columns on this subject and is largely responsible for the interest in it.[11] In 1986, the phrase was added to the Oxford English Dictionary with the earliest citation given as 1970.[11][26] The Historical Dictionary of American Slang (1997) cited Shepard's novel, thus pushing the earliest known usage back to 1967.[11]
Several key discoveries in further antedating the phrase were made by Bonnie Taylor-Blake, a neuroscience researcher at the University of North Carolina at Chapel Hill and an amateur member of the American Dialect Society, an association of professional and amateur linguists whose mailing list often serves as a forum for word and phrase discoveries. In 2012, Taylor-Blake discovered the 1956 and 1957 uses in Kentucky Happy Hunting Ground, and later that year she and Fred R. Shapiro found the "whole six yards" examples from the 1912–1921 period, which received substantial publicity.[11] In 2013, Taylor-Blake posted her discovery of the Mitchell Commercial uses from the 1907–1914 period.[27][28]
Origin[edit]
There is still no consensus on the origin, though many early published quotations are now available for study. A vast number of explanations for this phrase has been suggested;[29][30] many of these are no longer viable in light of what is now known about the phrase's history.
Perhaps the most commonly offered explanation is that World War II (1939–1945) aircraft machine gun belts were nine yards long. There are many versions of this explanation with variations regarding type of plane, nationality of gunner and geographic area. An alternative weapon is the ammunition belt for the British Vickers machine gun, invented and adopted by the British Army before World War I (1914–1918). The standard belt for this gun held 250 rounds of ammunition and was approximately twenty feet (under seven yards) in length. [11] However, the Vickers gun as fitted to aircraft during the First World War usually had ammunition containers capable of accommodating linked belts of 350-400 rounds, the average length of such a belt being about nine yards, and it was thought that this may be the origin of the phrase. [31] This theory is no longer considered viable, since the phrase predates World War I.
Another common explanation is that "nine yards" is a cubic measure and refers to the volume of a concrete mixer.[32] This theory, too, is inconsistent with the phrase's history.[11]
Many of the popular candidates relate to the length of pieces of fabric, or various garments, including Indian saris, Scottish kilts, burial shrouds, or bolts of cloth. No single source verifies that any one of those suggestions was the actual origin. However, an article published in Comments on Etymology demonstrates that fabric was routinely sold in standard lengths of nine yards (and other multiples of three yards) during the 1800s and early 1900s.[33] This may explain why so many different types of cloth or garments have been said to have been nine yards long.[34]The phrase "...she has put the whole nine yards into one shirt" appears in 1855. [35]
Other proposed sources include the volume of graves;[36] the length of bridal veils, kilts,[37] burial shrouds, bolts of cloth,[38] or saris; a very long list; ritual disembowelment; shipyards; and American football. Little documentary evidence has surfaced to support any of these explanations.[39]
One proposed origin involves the world of full-rigged sailing ships, in which yard is used not as a measure of length or size, but as the name of each horizontal spar on which a sail is hung. All square-rigged sails unfurled, with 3 yards on each of 3 masts, could then be described as the whole nine yards,[40][41] but again no actual documentation has been uncovered to support this explanation, and in any case not all ships had exactly three yards on each mast, even disregarding the fact that by no means all sailing vessels were three-masters. [42]
Bonnie Taylor-Blake, noting that several early examples are in the form "give" or "tell" the whole nine (or six) yards, has suggested that the idiom likely relied on "yards" as "lengthy or thorough presentation [of news, anecdotes, play-by-play, etc.]"[43]
Jesse Sheidlower, editor-at-large for the Oxford English Dictionary, and Fred R. Shapiro have argued that the phrase does not have a concrete meaning, pointing to the variance between six and nine yards and comparing it to the whole shebang.[11]
See also[edit]
Look up the whole nine yards in Wiktionary, the free dictionary.
The full monty (phrase)
Notes[edit]
Jump up ^ The phrase "to the nines" is first recorded in 1687. In early usage, it suggested that a work met the standards of the nine Muses of Greek mythology. Nine is considered a perfect number in numerology as it is three squared. Note that the phrase "cloud seven" was inflated to "cloud nine" by the same process. See The Phrase Finder and Random House Historical Dictionary of American Slang. Dressing "To the Nines" could also refer to the fact that in French "neuf" means both 9 and New ("Mes souliers sont neuf" = "My shoes are new"). So, "dressing to the nines" would mean dressing in a new outfit ("neuf").
References[edit]
Jump up ^ Whole, adj., n., and adv., C2, compound whole nine yards, Oxford English Dictionary (OED Online version Marc 2013) (citing Aviation Week 7 March 1983, 46/2).
Jump up ^ Shapiro, Fred, "You can quote them", May/June 2009
^ Jump up to: a b Nunberg, Geoff, "The Whole Nine Yards' Of What?", NPR Fresh Air, January 14, 2013.
Jump up ^ "Baseball", Mitchell Commercial, p. 2,col. 3 (May 2, 1907).
Jump up ^ Mitchell Commercial, p. 3, col. 5 (June 4, 1908).
Jump up ^ "Third Term Superstition", Mitchell Commercial p. 2, col. 2 (October 10, 1912).
Jump up ^ "Story of a Green Basket," Mitchell Commercial p. 1, col. 5 (November 26, 1914).
Jump up ^ "Livingston", Mount Vernon (Kentucky) Signal’, p. 1.
Jump up ^ "Mount Vernon Signal Newspaper" (PDF). Old Fulton NY Post Cards. June 28, 1912. Retrieved June 11, 2017.
Jump up ^ "The Whole Six Yards of It," Spartanburg (South Carolina) Herald-Journal, 7 May 1921, p. 5.
^ Jump up to: a b c d e f g h i Schuessler, Jennifer (December 26, 2012). "The Whole Nine Yards About a Phrase’s Origin". New York Times. Retrieved December 29, 2012. Then, in August, Bonnie Taylor-Blake, a neuroscience researcher in North Carolina who had been searching for variants of the phrase via Google News Archive and Google Books for five years, posted a message on the e-mail list of the American Dialect Society noting a 1956 occurrence in an outdoors magazine called Kentucky Happy Hunting Ground, followed in September by a more startling twist: a 1921 headline from The Spartanburg Herald-Journal in South Carolina reading "The Whole Six Yards of It." ... The recent discovery of several instances of "the whole six yards" in newspapers from the 1910s—four decades before the earliest known references to the whole nine yards—opens a new window onto "the most prominent etymological riddle of our time," said Fred Shapiro, a librarian at Yale Law School who announced the findings in next month’s issue of The Yale Alumni Magazine. ...
^ Jump up to: a b Zimmer, Ben (August 3, 2012). "Stretching Out 'The Whole Nine Yards'". Word Routes. Retrieved August 24, 2012.
Jump up ^ Rhody, Ron (July 1956). "Kentucky Afield Fishing Derbies Are Underway". Kentucky Happy Hunting Ground. 12 (4): 18.
Jump up ^ See also "Plugs and Bugs" by Ferd Holtmann, Kentucky Happy Hunting Ground, Vol. 18. No. 2, March, 1962, page 6. "Most anglers will tell you he has six or seven baits he uses more than all the others combined, yet he would be lost without the entire nine yards. It might be said there's a feeling of security involved in carrying the whole load on every outing."
Jump up ^ "The whole nine yards - meaning and origin", Gary Martin, 2006. The Phrase Finder
Jump up ^ Wegner, Robert E., "Man on the Thresh-Hold", Michigan's Voices, Fall 1962. "...the consequence of house, home, kids, respectability, status as a college professor and the whole nine yards, as a brush salesman who came by the house was fond of saying, the whole damn nine yards."
Jump up ^ Linster, Gale F., "Constructive Criticism", Car Life Vol. 9, Issue 11 (December 1962), p. 2. "Your staff of testers cannot fairly and equitably appraise the Chevrolet Impala sedan, with all nine yards of goodies, against the Plymouth Savoy which has straight shift and none of the mechanical conveniences which are quite common now."
Jump up ^ Trumbell, Stephen. "Talking Hip in the Space Age", Tucson Daily Citizen (Arizona), April 25, 1964; "'Give 'em the whole nine yards' means an item-by-item report on any project." (The reporter's name was misspelled in this newspaper; it is actually Trumbull.)
Zimmer, Benjamin (June 21, 2007). "Great moments in antedating". Language Log. University of Pennsylvania. Retrieved November 11, 2008.
Jump up ^ Zimmer, Ben (25 March 2009). "Where Did We Get "The Whole Nine Yards"?". Word Routes. Visual Thesaurus. Retrieved May 20, 2009.
Jump up ^ Campbell, Frank D., Jr. Lt. Col, Daily Facts, (Redlands, CA), April 7, 1965, p. 6. "We got the whole nine yards, including a side trip to Panama for jungle survival." (Quote attributed to Maj Clyde B. Williams.)
Burris, Keith, "'Burners' Are Not Informed", Deseret News, 28 December 1965, (Salt Lake City, Utah), p. A11, col. 1; "Capt. Greer was on alert the night of the big Red raid on Piel Me. He said the Cong troops were extremely well outfitted with steel helmets, boots -- 'the whole nine yards of uniform.'"
Jump up ^ Andrus, Col. Burton C. Jr, Assembly [magazine], Association of Graduates, United States Military Academy, v. 23 n. 3 col. 3, Fall 1965, p. 53 (55 of 100 in pdf). "We have 60 of the 120 rooms reserved so far--why not take over "the whole 9 yards" as the teenagers say?"
Jump up ^ Guthrie, James M., "Sesquicentennial Scrapbook", National Road Traveler, [Cambridge City, Indiana], June 30, 1966, p. 3; "If you like "The Old Swimmin' Hole," "Raggedy Man" and "When the Frost is on the Pumpkin" you'll like this one. And J.W. Riley is only a small contributor. (But Riley would have loved the whole nine yards)." For Guthrie's biographical information, see here.
Jump up ^ Technical Review, The Society of Experimental Test Pilots, Vol. 8, No. 2, p. 176 (September 1966) "Then two-engines, two pilots, and the rest, the nine yards of things that we have really all been aware of for a long time and should pay a lot more attention to."
Jump up ^ Krueger, Carl, Wings of the Tiger: A novel (1966); "'Okay, Tiger,' it said. 'Give 'em the whole nine yards. Now!'" [An instruction to fire at the enemy], p. 39. "We'll go over it after de-briefing. Get me a list of all pilots and planes available. Everything. The whole nine yards." p. 57.
Jump up ^ Shepard, Elaine, The Doom Pussy, Trident Press, (1967), p. 54; "Slipping out of the knot [marriage] was expensive but Smash was eventually able to untangle what he called 'the whole nine yards.'" The phrase appears several times in the book, always as the pet usage of Major "Smash" Crandell, a U.S. Air Force navigator. At one point, Smash refers to, "the ninth yard" (finishing touch).
Jump up ^ Whole, adj., n., and adv., C2, compound whole nine yards, Oxford English Dictionary (OED Online version March 2013).
Jump up ^ Bonnie Taylor-Blake, The whole nine yards (1908, 1912, 1914), ADS-L (September 6, 2013).
Jump up ^ Bonnie Taylor-Blake, The full nine yards (1907), ADS-L (September 6, 2013)..
Jump up ^ Wilton, Dave, "whole nine yards, the", June 21, 2007.
Jump up ^ "The whole 'whole nine yards' enchilada", The Phrase Finder. An amusing chart is included which shows which explanations are most common.
Jump up ^ Goldsmith, Dolf L (1994). The Grand Old Lady of No Man's Land, The Vickers Machinegun. Cobourg, Ontario, Canada: Collector Grade Publications Inc. p. 132. ISBN 0-88935-147-3.
Jump up ^ Wilton, David. Word Myths: Debunking Linguistic Urban Legends, Oxford University Press, 2005. ISBN 0-19-517284-1, p. 36. William Safire and James J. Kilpatrick have both given the cement mixer explanation.
Jump up ^ Reitan, Peter (January 2015). "Origin of The Whole Three/Six/Nine Yards: The Sale of Cloth in Multiples of Threes was Common in the 1800s and Early 1900s.". Comments on Etymology. 44 (4): 2.
Jump up ^ Brown, Peter Jensen. "Nine Yards to the Dollar - the History and Etymology of "the Whole Nine Yards"". Early Sports 'n' Pop-Culture History Blog. Retrieved 10 February 2015.
Jump up ^ Yankee Notions. T. W. Strong. 1855. p. 107.
Jump up ^ Wilton, p. 37. A grave is about 4 cubic yards.
Jump up ^ http://www.highlandsecrets.com/qa.html
Jump up ^ Wilton p. 36. A bolt of cloth is 20 to 25 yards.
Jump up ^ Wilton, pp. 34–38.
Jump up ^ Nautical Terms Index
Jump up ^ Nautical Language
Jump up ^ Wilton, p. 37. There was no standard number of yards on a sailboat, nor any citations of this phrase from the sailboat era.
Jump up ^ Bonnie Taylor-Blake, Front-page New York Times story on "The Whole Nine Yards", ADS-L (December 27, 2012).
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