Exponential Distribution - Yousef's Notes

A random variable $X$ is said to have an Exponential Distribution with parameter $\lambda \gt 0$ if its pdf is $$ \lambda e^{-\lambda x}, \quad \text{for } x \geq 0 $$

$P(X < x) = \int_{-\infty}^{x} f(x) , dx = 1 - e^{-\lambda x}$
$P(X > x) = e^{-\lambda x}$
$E(X) = \frac{1}{\lambda}$
$V(X) = \frac{1}{\lambda^2}$

The Exponential is (the only) Memoryless Random Variable:

$$ P(X > t + s \mid X > s) = \frac{P(X > t + s, X > s)}{P(X > s)} $$ $$ = \frac{P(X > t + s)}{P(X > s)} $$ $$ = \frac{e^{-\lambda(t+s)}}{e^{-\lambda s}} = e^{-\lambda t} $$ $$ = P(X > t) $$