Exponential Distribution

This might be related to the Geometric Distribution, as the continuous analogue

We say that $X$ is a random variable with exponential distribution with parameter $λ > 0$ , and we write $X \sim \exp (λ)$ , when the pdf is

f (x; λ) = {\begin{cases} λ e^{- λ x} & x > 0 \\ 0 & otherwise \end{cases}

We can calculate the cdf as

F (x; λ) = {\begin{cases} 1 - e^{- λ x} & x > 0 \\ 0 & otherwise \end{cases}

and that $P (X > x) = e^{- λ x}$

We have that

and the $n$ th moment of $X$ , we have that

E [X^{n}] = \frac{n!}{λ^{n}}

Let $c > 0$ be constant and $X$ be a random variable with distribution $\exp (λ)$ , we get that

c X \sim \exp (λ / c)

We can get the moment generating function of $X$

M (t) = \frac{λ}{λ - t} t < λ

The characteristic is $$ \phi(t) = \frac{\lambda}{\lambda-it} \qquad t < \lambda $$

Let $X$ be a random variable with exponential distribution with parameter $λ > 0$ . We get that for any $x, y \geq 0$ ,

P (X > x + y ∣ X > y) = P (X > x)

Let $U$ be a random variable with distribution $unif (0, 1)$ and let $λ > 0$ be a constant. Then the random variable $X$ , defined as

X = - \frac{1}{λ} \ln (1 - U)

Then $X \sim \exp (λ)$ .