Exponential distribution

An exponentially distributed random variable $𝑇 \sim E x p (𝜆)$ is common in situations of exponential decay, usually representing the amount of time taken for something to happen, e.g. the decay of a uranium nucleus (see Poisson process). It has positive support and is described by the following probability density function and Cumulative distribution function

𝑓 𝑇 (𝑡) = {𝜆 𝑒 - 𝜆 𝑡 𝑡 \geq 0 0 𝑡 < 0 𝐹 𝑇 (𝑡) = {1 - 𝑒 - 𝜆 𝑡 𝑡 \geq 0 0 𝑡 < 0

$𝜆$ may be thought of as the frequency of the event: the higher $𝜆$ , the shorter the time before the event occurs. In fact, the expected period is exactly what one might assume by this analogy.

Properties

Let $𝑇 \sim E x p (𝜆)$

Expectation: $𝔼 [𝑇] = 1 / 𝜆$
Variance: $V a r [𝑇] = 1 / 𝜆$
Moment-generating function: $𝑀 𝑇 (𝜏) = 𝜆 𝜆 - 𝜏$ for $𝜏 < 𝜆$
Moments: $𝔼 [𝑋 𝑛] = 𝑛! 𝜆 𝑛$

Additionally

Every positive, continuous Memoryless random variable has an exponential disrtribution.
Minimum of independent exponentially distributed random variables

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