= Exponential Distribution =

The '''exponential distribution''' is a non-negative continuous probability distribution function.

<<TableOfContents>>

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== Description ==

The distribution is parameterized to a strictly positive '''rate'''. A random variable distributed by this function is expressed as ''X ~ Exp(λ)''.

The distribution is given by {{attachment:exp.svg}} for any ''x ∈ [0,,∞)''.

The exponential distribution is equivalent to a [[Analysis/GammaDistribution|gamma distribution]] parameterized for shape and rate (''θ''): ''Exp(λ) = Gamma(1, λ)''.

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== Moments ==

The [[Analysis/ExpectedValue|expected value]] is given as ''E[X] = 1/λ''. The derivation is:

{{attachment:exp1.svg}}

[[Calculus/IntegrationByParts|Integrate by parts]] with ''u = x'' and ''dv = λe^-λx^ dx''.

{{attachment:exp2.svg}}

The first part of the RHS is an infinite [[Calculus/Limit|limit]] that evaluates as indeterminate, but applying [[Calculus/Limit#L.27H.2BAPQ-pital.27s_Rule|L'Hôpital's Rule]] reveals it to be 0.

{{attachment:exp3.svg}}

{{attachment:exp4.svg}}

Therefore the problem reduced back to a single integral.

{{attachment:exp5.svg}}

This is now also an infinite limit.

{{attachment:exp6.svg}}

This rewritten limit is clearly ''0 - 1 = -1'', leaving ''E[X] = 1/λ''.

[[Analysis/Variance|Variance]] is given as ''Var[X] = 1/(λ^2^)''.



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CategoryRicottone