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Exponential distribution
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<h2 id="definitions">Definitions</h2> <h3>Probability density function</h3> <p>The <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (pdf) of an exponential distribution is </p> 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}}} <p>Here <i>λ</i> > 0 is the parameter of the distribution, often called the <i>rate parameter</i>. The distribution is supported on the interval [0, ∞). If a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> <i>X</i> has this distribution, we write <i>X</i> ~ Exp(<i>λ</i>). </p><p>The exponential distribution exhibits <a href="/facts/Infinite_divisibility_(probability)/XsDLIfs2">infinite divisibility</a>. </p> <h3>Cumulative distribution function</h3> <p>The <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> is given by </p> 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}}} <h3>Alternative parametrization</h3> <p>The exponential distribution is sometimes parametrized in terms of the <a href="/facts/Scale_parameter/6lxcyKUf">scale parameter</a> <i>β</i> = 1/<i>λ</i>, which is also the mean: f ( x ; β ) = { 1 β e − x / β x ≥ 0 , 0 x < 0. F ( x ; β ) = { 1 − e − x / β x ≥ 0 , 0 x < 0. {\displaystyle f(x;\beta )={\begin{cases}{\frac {1}{\beta }}e^{-x/\beta }&x\geq 0,\\0&x<0.\end{cases}}\qquad \qquad F(x;\beta )={\begin{cases}1-e^{-x/\beta }&x\geq 0,\\0&x<0.\end{cases}}} </p> <h2 id="properties">Properties</h2> <h3>Mean, variance, moments, and median</h3> <p>The mean or <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of an exponentially distributed random variable <i>X</i> with rate parameter <i>λ</i> is given by E [ X ] = 1 λ . {\displaystyle \operatorname {E} [X]={\frac {1}{\lambda }}.} </p><p>In light of the examples given below, this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes. </p><p>The <a href="/facts/Variance/ULBJKXD1">variance</a> of <i>X</i> is given by Var [ X ] = 1 λ 2 , {\displaystyle \operatorname {Var} [X]={\frac {1}{\lambda ^{2}}},} so the <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> is equal to the mean. </p><p>The <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a> of <i>X</i>, 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}}}.} </p><p>The <a href="/facts/Central_moment/geXcMwyv">central moments</a> of <i>X</i>, 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 !<i>n</i> is the <a href="/facts/Subfactorial/1FnDzGrP">subfactorial</a> of <i>n</i> </p><p>The <a href="/facts/Median/YwXGWTyr">median</a> of <i>X</i> 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 <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>. Thus the <a href="/facts/Absolute_difference/2AdTdj8a">absolute difference</a> between the mean and median is | E [ X ] − m [ X ] | = 1 − ln ( 2 ) λ < 1 λ = σ [ X ] , {\displaystyle \left|\operatorname {E} \left[X\right]-\operatorname {m} \left[X\right]\right|={\frac {1-\ln(2)}{\lambda }}<{\frac {1}{\lambda }}=\operatorname {\sigma } [X],} </p><p>in accordance with the <a href="/facts/Median-mean_inequality/jf5ReDJv">median-mean inequality</a>. </p> <h3>Memorylessness property of exponential random variable</h3> <p>An exponentially distributed random variable <i>T</i> 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.} </p><p>This can be seen by considering the <a href="/facts/Complementary_cumulative_distribution_function/WaKU8tp4">complementary cumulative distribution function</a>: 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}}} </p><p>When <i>T</i> is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if <i>T</i> is conditioned on a failure to observe the event over some initial period of time <i>s</i>, 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 <a href="/facts/Conditional_probability/QcN2UERV">conditional probability</a> 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. </p><p>The exponential distribution and the <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a> are <a href="/facts/Memorylessness/8EgzaWtS">the only memoryless probability distributions</a>. </p><p>The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant <a href="/facts/Failure_rate/shlTIKKg">failure rate</a>. </p> <h3>Quantiles</h3> <p>The <a href="/facts/Quantile_function/sf0jbCXh">quantile function</a> (inverse cumulative distribution function) for Exp(<i>λ</i>) 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} </p><p>The <a href="/facts/Quartile/XLlNsqEP">quartiles</a> are therefore: </p> <ul><li>first quartile: ln(4/3)/<i>λ</i></li> <li><a href="/facts/Median/YwXGWTyr">median</a>: ln(2)/<i>λ</i></li> <li>third quartile: ln(4)/<i>λ</i></li></ul> <p>And as a consequence the <a href="/facts/Interquartile_range/tRk6u8w2">interquartile range</a> is ln(3)/<i>λ</i>. </p> <h3>Conditional Value at Risk (Expected Shortfall)</h3> <p>The conditional value at risk (CVaR) also known as the <a href="/facts/Expected_shortfall/03GNwfpV">expected shortfall</a> or superquantile for Exp(<i>λ</i>) is derived as follows:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p> q ¯ α ( X ) = 1 1 − α ∫ α 1 q p ( X ) d p = 1 ( 1 − α ) ∫ α 1 − ln ( 1 − p ) λ d p = − 1 λ ( 1 − α ) ∫ 1 − α 0 − ln ( y ) d y = − 1 λ ( 1 − α ) ∫ 0 1 − α ln ( y ) d y = − 1 λ ( 1 − α ) [ ( 1 − α ) ln ( 1 − α ) − ( 1 − α ) ] = − ln ( 1 − α ) + 1 λ {\displaystyle {\begin{aligned}{\bar {q}}_{\alpha }(X)&={\frac {1}{1-\alpha }}\int _{\alpha }^{1}q_{p}(X)dp\\&={\frac {1}{(1-\alpha )}}\int _{\alpha }^{1}{\frac {-\ln(1-p)}{\lambda }}dp\\&={\frac {-1}{\lambda (1-\alpha )}}\int _{1-\alpha }^{0}-\ln(y)dy\\&={\frac {-1}{\lambda (1-\alpha )}}\int _{0}^{1-\alpha }\ln(y)dy\\&={\frac {-1}{\lambda (1-\alpha )}}[(1-\alpha )\ln(1-\alpha )-(1-\alpha )]\\&={\frac {-\ln(1-\alpha )+1}{\lambda }}\\\end{aligned}}} </p> <h3>Buffered Probability of Exceedance (bPOE)</h3> <p class="note">Main article: Buffered probability of exceedance</p> <p>The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold x {\displaystyle x} . It is derived as follows:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p> p ¯ x ( X ) = { 1 − α | q ¯ α ( X ) = x } = { 1 − α | − ln ( 1 − α ) + 1 λ = x } = { 1 − α | ln ( 1 − α ) = 1 − λ x } = { 1 − α | e ln ( 1 − α ) = e 1 − λ x } = { 1 − α | 1 − α = e 1 − λ x } = e 1 − λ x {\displaystyle {\begin{aligned}{\bar {p}}_{x}(X)&=\{1-\alpha |{\bar {q}}_{\alpha }(X)=x\}\\&=\{1-\alpha |{\frac {-\ln(1-\alpha )+1}{\lambda }}=x\}\\&=\{1-\alpha |\ln(1-\alpha )=1-\lambda x\}\\&=\{1-\alpha |e^{\ln(1-\alpha )}=e^{1-\lambda x}\}=\{1-\alpha |1-\alpha =e^{1-\lambda x}\}=e^{1-\lambda x}\end{aligned}}} </p> <h3>Kullback–Leibler divergence</h3> <p>The directed <a href="/facts/Kullback%E2%80%93Leibler_divergence/nh7SjlPE">Kullback–Leibler divergence</a> in <a href="/facts/Nat_(unit)/UgNgfVgI">nats</a> 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}\parallel \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}}} </p> <h3>Maximum entropy distribution</h3> <p>Among all continuous probability distributions with <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> [0, ∞) and mean <i>μ</i>, the exponential distribution with <i>λ</i> = 1/<i>μ</i> has the largest <a href="/facts/Differential_entropy/EoJSyw95">differential entropy</a>. In other words, it is the <a href="/facts/Maximum_entropy_probability_distribution/MVwFzu49">maximum entropy probability distribution</a> for a <a href="/facts/Random_variate/dTkjAeaj">random variate</a> <i>X</i> which is greater than or equal to zero and for which E[<i>X</i>] is fixed.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h3>Distribution of the minimum of exponential random variables</h3> <p>Let <i>X</i>1, ..., <i>X</i><i>n</i> be <a href="/facts/Independent_random_variables/NUzQtnUL">independent</a> exponentially distributed random variables with rate parameters <i>λ</i>1, ..., <i>λn</i>. 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}.} </p><p>This can be seen by considering the <a href="/facts/Complementary_cumulative_distribution_function/WaKU8tp4">complementary cumulative distribution function</a>: 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 \left(X_{i}>x\right)\\={}&\prod _{i=1}^{n}\exp \left(-x\lambda _{i}\right)=\exp \left(-x\sum _{i=1}^{n}\lambda _{i}\right).\end{aligned}}} </p><p>The index of the variable which achieves the minimum is distributed according to the categorical distribution Pr ( X k = min { X 1 , … , X n } ) = λ k λ 1 + ⋯ + λ n . {\displaystyle \Pr \left(X_{k}=\min\{X_{1},\dotsc ,X_{n}\}\right)={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.} </p><p>A proof can be seen by letting I = argmin i ∈ { 1 , ⋯ , n } { X 1 , … , X n } {\displaystyle I=\operatorname {argmin} _{i\in \{1,\dotsb ,n\}}\{X_{1},\dotsc ,X_{n}\}} . Then, Pr ( I = k ) = ∫ 0 ∞ Pr ( X k = x ) Pr ( ∀ i ≠ k X i > x ) d x = ∫ 0 ∞ λ k e − λ k x ( ∏ i = 1 , i ≠ k n e − λ i x ) d x = λ k ∫ 0 ∞ e − ( λ 1 + ⋯ + λ n ) x d x = λ k λ 1 + ⋯ + λ n . {\displaystyle {\begin{aligned}\Pr(I=k)&=\int _{0}^{\infty }\Pr(X_{k}=x)\Pr(\forall _{i\neq k}X_{i}>x)\,dx\\&=\int _{0}^{\infty }\lambda _{k}e^{-\lambda _{k}x}\left(\prod _{i=1,i\neq k}^{n}e^{-\lambda _{i}x}\right)dx\\&=\lambda _{k}\int _{0}^{\infty }e^{-\left(\lambda _{1}+\dotsb +\lambda _{n}\right)x}dx\\&={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.\end{aligned}}} </p><p>Note that max { X 1 , … , X n } {\displaystyle \max\{X_{1},\dotsc ,X_{n}\}} is not exponentially distributed, if <i>X</i>1, ..., <i>X</i><i>n</i> do not all have parameter 0.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h3>Joint moments of i.i.d. exponential order statistics</h3> <p>Let X 1 , … , X n {\displaystyle X_{1},\dotsc ,X_{n}} be n {\displaystyle n} <a href="/facts/Independent_and_identically_distributed/othIRaWt">independent and identically distributed</a> exponential random variables with rate parameter <i>λ</i>. Let X ( 1 ) , … , X ( n ) {\displaystyle X_{(1)},\dotsc ,X_{(n)}} denote the corresponding <a href="/facts/Order_statistic/QS3RnGcC">order statistics</a>. For i < j {\displaystyle i<j} , the joint moment E [ X ( i ) X ( j ) ] {\displaystyle \operatorname {E} \left[X_{(i)}X_{(j)}\right]} of the order statistics X ( i ) {\displaystyle X_{(i)}} and X ( j ) {\displaystyle X_{(j)}} is given by E [ X ( i ) X ( j ) ] = ∑ k = 0 j − 1 1 ( n − k ) λ E [ X ( i ) ] + E [ X ( i ) 2 ] = ∑ k = 0 j − 1 1 ( n − k ) λ ∑ k = 0 i − 1 1 ( n − k ) λ + ∑ k = 0 i − 1 1 ( ( n − k ) λ ) 2 + ( ∑ k = 0 i − 1 1 ( n − k ) λ ) 2 . {\displaystyle {\begin{aligned}\operatorname {E} \left[X_{(i)}X_{(j)}\right]&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\operatorname {E} \left[X_{(i)}\right]+\operatorname {E} \left[X_{(i)}^{2}\right]\\&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\sum _{k=0}^{i-1}{\frac {1}{(n-k)\lambda }}+\sum _{k=0}^{i-1}{\frac {1}{((n-k)\lambda )^{2}}}+\left(\sum _{k=0}^{i-1}{\frac {1}{(n-k)\lambda }}\right)^{2}.\end{aligned}}} </p><p>This can be seen by invoking the <a href="/facts/Law_of_total_expectation/zSqn8uVj">law of total expectation</a> 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 [ E [ X ( j ) ] + x ] f X ( i ) ( x ) d x ( by the memoryless property ) = ∑ k = 0 j − 1 1 ( n − k ) λ E [ X ( i ) ] + E [ X ( i ) 2 ] . {\displaystyle {\begin{aligned}\operatorname {E} \left[X_{(i)}X_{(j)}\right]&=\int _{0}^{\infty }\operatorname {E} \left[X_{(i)}X_{(j)}\mid X_{(i)}=x\right]f_{X_{(i)}}(x)\,dx\\&=\int _{x=0}^{\infty }x\operatorname {E} \left[X_{(j)}\mid X_{(j)}\geq x\right]f_{X_{(i)}}(x)\,dx&&\left({\textrm {since}}~X_{(i)}=x\implies X_{(j)}\geq x\right)\\&=\int _{x=0}^{\infty }x\left[\operatorname {E} \left[X_{(j)}\right]+x\right]f_{X_{(i)}}(x)\,dx&&\left({\text{by the memoryless property}}\right)\\&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\operatorname {E} \left[X_{(i)}\right]+\operatorname {E} \left[X_{(i)}^{2}\right].\end{aligned}}} </p><p>The first equation follows from the <a href="/facts/Law_of_total_expectation/zSqn8uVj">law of total expectation</a>. 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} \left[X_{(j)}\mid X_{(j)}\geq x\right]} with E [ X ( j ) ] + x {\displaystyle \operatorname {E} \left[X_{(j)}\right]+x} . </p> <h3>Sum of two independent exponential random variables</h3> <p>The probability distribution function (PDF) of a sum of two independent random variables is the <a href="/facts/Convolution_of_probability_distributions/l467HgVK">convolution of their individual PDFs</a>. If X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are independent exponential random variables 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 ) = ∫ − ∞ ∞ f X 1 ( x 1 ) f X 2 ( z − x 1 ) d x 1 = ∫ 0 z λ 1 e − λ 1 x 1 λ 2 e − λ 2 ( z − x 1 ) d x 1 = λ 1 λ 2 e − λ 2 z ∫ 0 z e ( λ 2 − λ 1 ) x 1 d x 1 = { λ 1 λ 2 λ 2 − λ 1 ( e − λ 1 z − e − λ 2 z ) if λ 1 ≠ λ 2 λ 2 z e − λ z if λ 1 = λ 2 = λ . {\displaystyle {\begin{aligned}f_{Z}(z)&=\int _{-\infty }^{\infty }f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})\,dx_{1}\\&=\int _{0}^{z}\lambda _{1}e^{-\lambda _{1}x_{1}}\lambda _{2}e^{-\lambda _{2}(z-x_{1})}\,dx_{1}\\&=\lambda _{1}\lambda _{2}e^{-\lambda _{2}z}\int _{0}^{z}e^{(\lambda _{2}-\lambda _{1})x_{1}}\,dx_{1}\\&={\begin{cases}{\dfrac {\lambda _{1}\lambda _{2}}{\lambda _{2}-\lambda _{1}}}\left(e^{-\lambda _{1}z}-e^{-\lambda _{2}z}\right)&{\text{ if }}\lambda _{1}\neq \lambda _{2}\\[4pt]\lambda ^{2}ze^{-\lambda z}&{\text{ if }}\lambda _{1}=\lambda _{2}=\lambda .\end{cases}}\end{aligned}}} The entropy of this distribution is available in closed form: assuming λ 1 > λ 2 {\displaystyle \lambda _{1}>\lambda _{2}} (without loss of generality), then H ( Z ) = 1 + γ + ln ( λ 1 − λ 2 λ 1 λ 2 ) + ψ ( λ 1 λ 1 − λ 2 ) , {\displaystyle {\begin{aligned}H(Z)&=1+\gamma +\ln \left({\frac {\lambda _{1}-\lambda _{2}}{\lambda _{1}\lambda _{2}}}\right)+\psi \left({\frac {\lambda _{1}}{\lambda _{1}-\lambda _{2}}}\right),\end{aligned}}} where γ {\displaystyle \gamma } is the <a href="/facts/Euler-Mascheroni_constant/2kEgRvjC">Euler-Mascheroni constant</a>, and ψ ( ⋅ ) {\displaystyle \psi (\cdot )} is the <a href="/facts/Digamma_function/nDTNu86e">digamma function</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>In the case of equal rate parameters, the result is an <a href="/facts/Erlang_distribution/dqFqlxJg">Erlang distribution</a> with shape 2 and parameter λ , {\displaystyle \lambda ,} which in turn is a special case of <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a>. </p><p>The sum of n independent Exp(<i>λ)</i> exponential random variables is Gamma(n, <i>λ)</i> distributed. </p> <h2 id="related-distributions">Related distributions</h2> <ul><li>If <i>X</i> ~ <a href="/facts/Laplace_distribution/teMwaLry">Laplace(μ, β−1)</a>, then |<i>X</i> − μ| ~ Exp(β).<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li> <li>If <i>X</i> ~ <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT"><i>U</i>(0, 1)</a> then −log(<i>X</i>) ~ Exp(1).</li> <li>If <i>X</i> ~ <a href="/facts/Pareto_distribution/PgMDbLrR">Pareto(1, λ)</a>, then log(<i>X</i>) ~ Exp(λ).<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li> <li>If <i>X</i> ~ <a href="/facts/Skew-logistic_distribution/bH0v6w4n">SkewLogistic(θ)</a>, then log ( 1 + e − X ) ∼ Exp ( θ ) {\displaystyle \log \left(1+e^{-X}\right)\sim \operatorname {Exp} (\theta )} .</li> <li>If <i>Xi</i> ~ <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT"><i>U</i>(0, 1)</a> 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 \operatorname {Exp} (1)} </li> <li>The exponential distribution is a limit of a scaled <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>: lim n → ∞ n Beta ( 1 , n ) = Exp ( 1 ) . {\displaystyle \lim _{n\to \infty }n\operatorname {Beta} (1,n)=\operatorname {Exp} (1).} </li> <li>The exponential distribution is a special case of type 3 <a href="/facts/Pearson_distribution/L0oULaWK">Pearson distribution</a>.</li> <li>The exponential distribution is the special case of a <a href="/facts/Gamma_distribution/lczcdJmw">Gamma distribution</a> with shape parameter 1.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li> <li>If <i>X</i> ~ Exp(λ) and <i>X</i><i>i</i> ~ Exp(λ<i>i</i>) then: <ul><li> k X ∼ Exp ( λ k ) {\displaystyle kX\sim \operatorname {Exp} \left({\frac {\lambda }{k}}\right)} , closure under scaling by a positive factor.</li> <li>1 + <i>X</i> ~ <a href="/facts/Benktander_Weibull_distribution/xwUKau8c">BenktanderWeibull</a>(λ, 1), which reduces to a truncated exponential distribution.</li> <li><i>keX</i> ~ <a href="/facts/Pareto_distribution/PgMDbLrR">Pareto</a>(<i>k</i>, λ).<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li> <li><i>e−λX</i> ~ <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT"><i>U</i>(0, 1)</a>.</li> <li><i>e−X</i> ~ <a href="/facts/Beta_distribution/DbJk4eeV">Beta</a>(λ, 1).<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a></li> <li>1/k<i>e</i><i>X</i> ~ <a href="/facts/Power_law/DhAjYwwz">PowerLaw</a>(<i>k</i>, λ)</li> <li> X ∼ Rayleigh ( 1 2 λ ) {\displaystyle {\sqrt {X}}\sim \operatorname {Rayleigh} \left({\frac {1}{\sqrt {2\lambda }}}\right)} , the <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li> <li> X ∼ Weibull ( 1 λ , 1 ) {\displaystyle X\sim \operatorname {Weibull} \left({\frac {1}{\lambda }},1\right)} , the <a href="/facts/Weibull_distribution/1fEWxbKl">Weibull distribution</a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></li> <li> X 2 ∼ Weibull ( 1 λ 2 , 1 2 ) {\displaystyle X^{2}\sim \operatorname {Weibull} \left({\frac {1}{\lambda ^{2}}},{\frac {1}{2}}\right)} <a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a></li> <li>μ − β log(λ<i>X</i>) ∼ <a href="/facts/Gumbel_distribution/PGkDYzwe">Gumbel</a>(μ, β).</li> <li> ⌊ X ⌋ ∼ Geometric ( 1 − e − λ ) {\displaystyle \lfloor X\rfloor \sim \operatorname {Geometric} \left(1-e^{-\lambda }\right)} , a <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a> on 0,1,2,3,...</li> <li> ⌈ X ⌉ ∼ Geometric ( 1 − e − λ ) {\displaystyle \lceil X\rceil \sim \operatorname {Geometric} \left(1-e^{-\lambda }\right)} , a <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a> on 1,2,3,4,...</li> <li>If also <i>Y</i> ~ Erlang(<i>n</i>, λ) or Y ∼ Γ ( n , 1 λ ) {\displaystyle Y\sim \Gamma \left(n,{\frac {1}{\lambda }}\right)} then X Y + 1 ∼ Pareto ( 1 , n ) {\displaystyle {\frac {X}{Y}}+1\sim \operatorname {Pareto} (1,n)} </li> <li>If also λ ~ <a href="/facts/Gamma_distribution/lczcdJmw">Gamma</a>(<i>k</i>, θ) (shape, scale parametrisation) then the marginal distribution of <i>X</i> is <a href="/facts/Lomax_distribution/ch9VY1mn">Lomax</a>(<i>k</i>, 1/θ), the gamma <a href="/facts/Compound_distribution/ccJIMBxT">mixture</a></li> <li>λ1<i>X</i>1 − λ2<i>Y</i>2 ~ <a href="/facts/Laplace_distribution/teMwaLry">Laplace(0, 1)</a>.</li> <li>min{<i>X</i>1, ..., <i>Xn</i>} ~ Exp(λ1 + ... + λ<i>n</i>).</li> <li>If also λ<i>i</i> = λ then: <ul><li> X 1 + ⋯ + X k = ∑ i X i ∼ {\displaystyle X_{1}+\cdots +X_{k}=\sum _{i}X_{i}\sim } <a href="/facts/Erlang_distribution/dqFqlxJg">Erlang</a>(<i>k</i>, λ) = <a href="/facts/Gamma_distribution/lczcdJmw">Gamma</a>(<i>k</i>, λ) with integer shape parameter <i>k</i> and rate parameter λ.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></li> <li>If T = ( X 1 + ⋯ + X n ) = ∑ i = 1 n X i {\displaystyle T=(X_{1}+\cdots +X_{n})=\sum _{i=1}^{n}X_{i}} , then 2 λ T ∼ χ 2 n 2 {\displaystyle 2\lambda T\sim \chi _{2n}^{2}} .</li> <li><i>X</i><i>i</i> − <i>X</i><i>j</i> ~ Laplace(0, λ−1).</li></ul></li> <li>If also <i>X</i><i>i</i> are independent, then: <ul><li> X i X i + X j {\displaystyle {\frac {X_{i}}{X_{i}+X_{j}}}} ~ <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">U</a>(0, 1)</li> <li> Z = λ i X i λ j X j {\displaystyle Z={\frac {\lambda _{i}X_{i}}{\lambda _{j}X_{j}}}} 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 <a href="/facts/Confidence_interval/NS8mG5UE">confidence interval</a> for λ i λ j {\displaystyle {\frac {\lambda _{i}}{\lambda _{j}}}} .</li></ul></li> <li>If also λ = 1: <ul><li> μ − β log ( e − X 1 − e − X ) ∼ Logistic ( μ , β ) {\displaystyle \mu -\beta \log \left({\frac {e^{-X}}{1-e^{-X}}}\right)\sim \operatorname {Logistic} (\mu ,\beta )} , the <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distribution</a></li> <li> μ − β log ( X i X j ) ∼ Logistic ( μ , β ) {\displaystyle \mu -\beta \log \left({\frac {X_{i}}{X_{j}}}\right)\sim \operatorname {Logistic} (\mu ,\beta )} </li> <li><i>μ</i> − σ log(<i>X</i>) ~ <a href="/facts/Generalized_extreme_value_distribution/hDG41s9y">GEV(μ, σ, 0)</a>.</li> <li>Further if Y ∼ Γ ( α , β α ) {\displaystyle Y\sim \Gamma \left(\alpha ,{\frac {\beta }{\alpha }}\right)} then X Y ∼ K ( α , β ) {\displaystyle {\sqrt {XY}}\sim \operatorname {K} (\alpha ,\beta )} (<a href="/facts/K-distribution/KHnP6UTg">K-distribution</a>)</li></ul></li> <li>If also λ = 1/2 then <i>X</i> ∼ χ22; i.e., <i>X</i> has a <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared distribution</a> with 2 <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86">degrees of freedom</a>. 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} (\lambda )={\frac {1}{2\lambda }}\operatorname {Exp} \left({\frac {1}{2}}\right)\sim {\frac {1}{2\lambda }}\chi _{2}^{2}\Rightarrow \sum _{i=1}^{n}\operatorname {Exp} (\lambda )\sim {\frac {1}{2\lambda }}\chi _{2n}^{2}} </li></ul></li> <li>If X ∼ Exp ( 1 λ ) {\displaystyle X\sim \operatorname {Exp} \left({\frac {1}{\lambda }}\right)} and Y ∣ X {\displaystyle Y\mid X} ~ <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson(<i>X</i>)</a> then Y ∼ Geometric ( 1 1 + λ ) {\displaystyle Y\sim \operatorname {Geometric} \left({\frac {1}{1+\lambda }}\right)} (<a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a>)</li> <li>The <a href="/facts/Hoyt_distribution/HoyxF3AF">Hoyt distribution</a> can be obtained from exponential distribution and <a href="/facts/Arcsine_distribution/KBWLLS3k">arcsine distribution</a></li> <li>The exponential distribution is a limit of the <a href="/facts/Kaniadakis_Exponential_distribution/r1x1khCr"><i>κ</i>-exponential distribution</a> in the κ = 0 {\displaystyle \kappa =0} case.</li> <li>Exponential distribution is a limit of the κ-Generalized Gamma distribution in the α = 1 {\displaystyle \alpha =1} and ν = 1 {\displaystyle \nu =1} cases: lim ( α , ν ) → ( 0 , 1 ) p κ ( x ) = ( 1 + κ ν ) ( 2 κ ) ν Γ ( 1 2 κ + ν 2 ) Γ ( 1 2 κ − ν 2 ) α λ ν Γ ( ν ) x α ν − 1 exp κ ( − λ x α ) = λ e − λ x {\displaystyle \lim _{(\alpha ,\nu )\to (0,1)}p_{\kappa }(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\Big )}}}{\frac {\alpha \lambda ^{\nu }}{\Gamma (\nu )}}x^{\alpha \nu -1}\exp _{\kappa }(-\lambda x^{\alpha })=\lambda e^{-\lambda x}} </li></ul> <p>Other related distributions: </p> <ul><li><a href="/facts/Hyper-exponential_distribution/vxa9JHSY">Hyper-exponential distribution</a> – the distribution whose density is a weighted sum of exponential densities.</li> <li><a href="/facts/Hypoexponential_distribution/hFDgbohC">Hypoexponential distribution</a> – the distribution of a general sum of exponential random variables.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li><a href="/facts/ExGaussian_distribution/mJU9306M">exGaussian distribution</a> – the sum of an exponential distribution and a <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a>.</li></ul> <h2 id="statistical-inference">Statistical inference</h2> <p>Below, suppose random variable <i>X</i> is exponentially distributed with rate parameter λ, and x 1 , … , x n {\displaystyle x_{1},\dotsc ,x_{n}} are <i>n</i> independent samples from <i>X</i>, with sample mean x ¯ {\displaystyle {\bar {x}}} . </p> <h3>Parameter estimation</h3> <p>The <a href="/facts/Maximum_likelihood/0Yq2dpQD">maximum likelihood</a> estimator for λ is constructed as follows. </p><p>The <a href="/facts/Likelihood_function/1L8P4NBX">likelihood function</a> for λ, given an <a href="/facts/Independent_and_identically_distributed/othIRaWt">independent and identically distributed</a> sample <i>x</i> = (<i>x</i>1, ..., <i>x</i><i>n</i>) 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),} </p><p>where: x ¯ = 1 n ∑ i = 1 n x i {\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}} is the sample mean. </p><p>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 {d}{d\lambda }}\ln L(\lambda )={\frac {d}{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}}} </p><p>Consequently, the <a href="/facts/Maximum_likelihood/0Yq2dpQD">maximum likelihood</a> estimate for the rate parameter is: λ ^ mle = 1 x ¯ = n ∑ i x i {\displaystyle {\widehat {\lambda }}_{\text{mle}}={\frac {1}{\overline {x}}}={\frac {n}{\sum _{i}x_{i}}}} </p><p>This is not an <a href="/facts/Unbiased_estimator/oxIvEgmd">unbiased estimator</a> of λ , {\displaystyle \lambda ,} although x ¯ {\displaystyle {\overline {x}}} is an unbiased<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> MLE<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> estimator of 1 / λ {\displaystyle 1/\lambda } and the distribution mean. </p><p>The bias of λ ^ mle {\displaystyle {\widehat {\lambda }}_{\text{mle}}} is equal to B ≡ E [ ( λ ^ mle − λ ) ] = λ n − 1 {\displaystyle B\equiv \operatorname {E} \left[\left({\widehat {\lambda }}_{\text{mle}}-\lambda \right)\right]={\frac {\lambda }{n-1}}} which yields the <a href="/facts/Maximum_likelihood_estimation/0Yq2dpQD">bias-corrected maximum likelihood estimator</a> λ ^ mle ∗ = λ ^ mle − B . {\displaystyle {\widehat {\lambda }}_{\text{mle}}^{*}={\widehat {\lambda }}_{\text{mle}}-B.} </p><p>An approximate minimizer of <a href="/facts/Mean_squared_error/kz3TR7bv">mean squared error</a> (see also: <a href="/facts/Bias%E2%80%93variance_tradeoff/xqDnRpeQ">bias–variance tradeoff</a>) can be found, assuming a sample size greater than two, with a correction factor to the MLE: 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 <a href="/facts/Inverse-gamma_distribution/zpyktNyN">inverse-gamma distribution</a>, Inv-Gamma ( n , λ ) {\textstyle {\mbox{Inv-Gamma}}(n,\lambda )} .<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> </p> <h3>Fisher information</h3> <p>The <a href="/facts/Fisher_information/Q2JLexN9">Fisher information</a>, 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} </p><p>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}.} </p><p>This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter λ {\displaystyle \lambda } . </p> <h3>Confidence intervals</h3> <p>An exact 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by:<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> 2 n λ ^ mle χ α 2 , 2 n 2 < 1 λ < 2 n λ ^ mle χ 1 − α 2 , 2 n 2 , {\displaystyle {\frac {2n}{{\widehat {\lambda }}_{\textrm {mle}}\chi _{{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n}{{\widehat {\lambda }}_{\textrm {mle}}\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<i>p</i>,<i>v</i> is the 100(<i>p</i>) <a href="/facts/Percentile/YKD2SDsw">percentile</a> of the <a href="/facts/Chi_squared_distribution/wYnWWgUe">chi squared distribution</a> with <i>v</i> <a href="/facts/Degrees_of_freedom_(statistics)/fvwKCU86">degrees of freedom</a>, n is the number of observations and x-bar is the sample average. A simple approximation to the exact interval endpoints can be derived using a normal approximation to the <i>χ</i>2<i>p</i>,<i>v</i> distribution. This approximation gives the following values for a 95% confidence interval: λ lower = λ ^ ( 1 − 1.96 n ) λ upper = λ ^ ( 1 + 1.96 n ) {\displaystyle {\begin{aligned}\lambda _{\text{lower}}&={\widehat {\lambda }}\left(1-{\frac {1.96}{\sqrt {n}}}\right)\\\lambda _{\text{upper}}&={\widehat {\lambda }}\left(1+{\frac {1.96}{\sqrt {n}}}\right)\end{aligned}}} </p><p>This approximation may be acceptable for samples containing at least 15 to 20 elements.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> </p> <h3>Bayesian inference</h3> <p>The <a href="/facts/Conjugate_prior/ScCFcs8b">conjugate prior</a> for the exponential distribution is the <a href="/facts/Gamma_distribution/lczcdJmw">gamma distribution</a> (of which the exponential distribution is a special case). The following parameterization of the gamma probability density function is useful: </p><p> 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 ).} </p><p>The <a href="/facts/Posterior_distribution/fojsMD0D">posterior distribution</a> <i>p</i> can then be expressed in terms of the likelihood function defined above and a gamma prior: </p><p> p ( λ ) ∝ L ( λ ) Γ ( 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 )\Gamma (\lambda ;\alpha ,\beta )\\&=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right){\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}}} </p><p>Now the posterior density <i>p</i> 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><p> p ( λ ) = Gamma ( λ ; α + n , β + n x ¯ ) . {\displaystyle p(\lambda )=\operatorname {Gamma} (\lambda ;\alpha +n,\beta +n{\overline {x}}).} </p><p>Here the <a href="/facts/Hyperparameter_(Bayesian_statistics)/zqcLPQcb">hyperparameter</a> <i>α</i> can be interpreted as the number of prior observations, and <i>β</i> 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}}}}.} </p> <h2 id="occurrence-and-applications">Occurrence and applications</h2> <h3>Occurrence of events</h3> <p>The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a>. </p><p>The exponential distribution may be viewed as a continuous counterpart of the <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a>, which describes the number of <a href="/facts/Bernoulli_trial/7fm8snCf">Bernoulli trials</a> necessary for a <i>discrete</i> process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state. </p><p>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: </p> <ul><li>The time until a radioactive <a href="/facts/Particle_decay/3koAz34P">particle decays</a>, or the time between clicks of a <a href="/facts/Geiger_counter/3xDNYh7I">Geiger counter</a></li> <li>The time between receiving one telephone call and the next</li> <li>The time until default (on payment to company debt holders) in reduced-form credit risk modeling</li></ul> <p>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 <a href="/facts/Mutation/Ee4NnWbY">mutations</a> on a <a href="/facts/DNA/nB89Iauz">DNA</a> strand, or between <a href="/facts/Roadkill/lxl6xVRu">roadkills</a> on a given road. </p><p>In <a href="/facts/Queuing_theory/8NDl56EM">queuing theory</a>, 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 <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> 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 <a href="/facts/Erlang_distribution/dqFqlxJg">Erlang distribution</a> (which is the distribution of the sum of several independent exponentially distributed variables). <a href="/facts/Reliability_theory/cd6AS5w3">Reliability theory</a> and <a href="/facts/Reliability_engineering/cd6AS5w3">reliability engineering</a> also make extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constant <a href="/facts/Hazard_rate/q8w13cjl">hazard rate</a> portion of the <a href="/facts/Bathtub_curve/sfPGsn9e">bathtub curve</a> used in reliability theory. It is also very convenient because it is so easy to add <a href="/facts/Failure_rate/shlTIKKg">failure rates</a> 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. </p> <p>In <a href="/facts/Physics/a7aH2y08">physics</a>, if you observe a <a href="/facts/Gas/hVVAoodV">gas</a> at a fixed <a href="/facts/Temperature/QyGe4gmj">temperature</a> and <a href="/facts/Pressure/eyc37GRG">pressure</a> in a uniform <a href="/facts/Gravitational_field/K1uTmJFy">gravitational field</a>, the heights of the various molecules also follow an approximate exponential distribution, known as the <a href="/facts/Barometric_formula/1J3vAqMc">Barometric formula</a>. This is a consequence of the entropy property mentioned below. </p><p>In <a href="/facts/Hydrology/Xh727cnE">hydrology</a>, 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.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> The blue picture illustrates an example of fitting the exponential distribution to ranked annually maximum one-day rainfalls showing also the 90% <a href="/facts/Confidence_belt/NS8mG5UE">confidence belt</a> based on the <a href="/facts/Binomial_distribution/UMoFMjDj">binomial distribution</a>. The rainfall data are represented by <a href="/facts/Plotting_position/h2jKBjYm">plotting positions</a> as part of the <a href="/facts/Cumulative_frequency_analysis/NtFhHr9h">cumulative frequency analysis</a>. <p>In operating-rooms management, the distribution of surgery duration for a category of surgeries with <a href="/facts/Predictive_methods_for_surgery_duration/H7QD4ASz">no typical work-content</a> (like in an emergency room, encompassing all types of surgeries). </p> <h3>Prediction</h3> <p>Having observed a sample of <i>n</i> 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 <i>λ</i> 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 <i>x</i><i>n</i>+1, conditioned on the observed samples <i>x</i> = (<i>x</i>1, ..., <i>xn</i>) 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).} </p><p>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. </p><p>A predictive distribution free of the issues of choosing priors that arise under the subjective Bayesian approach is </p><p> 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}}},} </p><p>which can be considered as </p> <ol><li>a frequentist <a href="/facts/Confidence_distribution/eGdL8uo7">confidence distribution</a>, obtained from the distribution of the pivotal quantity x n + 1 / x ¯ {\displaystyle {x_{n+1}}/{\overline {x}}} ;<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a></li> <li>a profile predictive likelihood, obtained by eliminating the parameter <i>λ</i> from the joint likelihood of <i>x</i><i>n</i>+1 and <i>λ</i> by maximization;<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a></li> <li>an objective Bayesian predictive posterior distribution, obtained using the non-informative <a href="/facts/Jeffreys_prior/6j4Wpu7t">Jeffreys prior</a> 1/<i>λ</i>;</li> <li>the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations.<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a></li></ol> <p>The accuracy of a predictive distribution may be measured using the distance or divergence between the true exponential distribution with rate parameter, <i>λ</i>0, and the predictive distribution based on the sample <i>x</i>. The <a href="/facts/Kullback%E2%80%93Leibler_divergence/nh7SjlPE">Kullback–Leibler divergence</a> is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(<i>λ</i>0||<i>p</i>) denote the Kullback–Leibler divergence between an exponential with rate parameter <i>λ</i>0 and a predictive distribution <i>p</i> it can be shown that </p><p> 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}\operatorname {E} _{\lambda _{0}}\left[\Delta (\lambda _{0}\parallel p_{\rm {ML}})\right]&=\psi (n)+{\frac {1}{n-1}}-\log(n)\\\operatorname {E} _{\lambda _{0}}\left[\Delta (\lambda _{0}\parallel p_{\rm {CNML}})\right]&=\psi (n)+{\frac {1}{n}}-\log(n)\end{aligned}}} </p><p>where the expectation is taken with respect to the exponential distribution with rate parameter <i>λ</i>0 ∈ (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 <i>n</i> > 0. </p> <h2 id="random-variate-generation">Random variate generation</h2> <p class="note">Further information: <a href="/facts/Non-uniform_random_variate_generation/xIkyQJqu">Non-uniform random variate generation</a></p> <p>A conceptually very simple method for generating exponential <a href="/facts/Variate/dTkjAeaj">variates</a> is based on <a href="/facts/Inverse_transform_sampling/fIDxBxT6">inverse transform sampling</a>: Given a random variate <i>U</i> drawn from the <a href="/facts/Uniform_distribution_(continuous)/XbnlVljT">uniform distribution</a> on the unit interval (0, 1), the variate </p><p> T = F − 1 ( U ) {\displaystyle T=F^{-1}(U)} </p><p>has an exponential distribution, where <i>F</i>−1 is the <a href="/facts/Quantile_function/sf0jbCXh">quantile function</a>, defined by </p><p> F − 1 ( p ) = − ln ( 1 − p ) λ . {\displaystyle F^{-1}(p)={\frac {-\ln(1-p)}{\lambda }}.} </p><p>Moreover, if <i>U</i> is uniform on (0, 1), then so is 1 − <i>U</i>. This means one can generate exponential variates as follows: </p><p> T = − ln ( U ) λ . {\displaystyle T={\frac {-\ln(U)}{\lambda }}.} </p><p>Other methods for generating exponential variates are discussed by Knuth<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> and Devroye.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> </p><p>A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dead_time/KYCGJ5cX">Dead time</a> – an application of exponential distribution to particle detector analysis.</li> <li><a href="/facts/Laplace_distribution/teMwaLry">Laplace distribution</a>, or the "double exponential distribution".</li> <li><a href="/facts/Relationships_among_probability_distributions/2aS20E3h">Relationships among probability distributions</a></li> <li><a href="/facts/Marshall%E2%80%93Olkin_exponential_distribution/MRX3Mrdc">Marshall–Olkin exponential distribution</a></li></ul> <h2 id="external-links">External links</h2> The Wikibook <i>Probability</i> has a page on the topic of: <i>Exponential distribution</i> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Exponential_distribution">"Exponential distribution"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="http://www.elektro-energetika.cz/calculations/ex.php?language=english">Online calculator of Exponential Distribution</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"7.2: Exponential Distribution". 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