The **exponential distribution** with rate is defined by

Since is differentiable, this is a continuous probability distribution, with density

One of the main reason this distribution is interesting is the so-called memoryless property:

Theorem 1If and , then

*Proof:* We compute

To compute the moments of the exponential distribution, we will make use of the following lemma:

Theorem 2If , then for any integer ,

where is the distribution function of .

*Proof:* By the Fubini-Tonelli theorem, we have

from which the result follows.

We now need to compute

We show by induction that this is equal to . For , we have

Assume the claim is true for some nonnegative integer , then

It follows then that

for . To verify, let’s use another method. The **moment-generating function** of a random variable is

for all such that this expectation exists. Since

by nonnegativity and monotonicity of the above series, we may use the monotone convergence theorem to compute

Since this series converges absolutely, for any positive integer ,

Setting , all terms except for are zero, and hence

The moment-generating function of the exponential distribution is

Hence

so that