Exponential Distribution
The exponential distribution is a non-negative continuous probability distribution function.
Contents
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 
for any x ∈ [0,,∞).
The exponential distribution is equivalent to a gamma distribution parameterized for shape and rate (θ): Exp(λ) = Gamma(1, λ).
Moments
The expected value is given as E[X] = 1/λ. The derivation is:
Integrate by parts with u = x and dv = λe-λx dx.
The first part of the RHS is an infinite limit that evaluates as indeterminate, but applying L'Hôpital's Rule reveals it to be 0.
Therefore the problem reduced back to a single integral.
This is now also an infinite limit.
This rewritten limit is clearly 0 - 1 = -1, leaving E[X] = 1/λ.
Variance is given as Var[X] = 1/(λ2).