Exponential Distribution

Exponential Distribution (Definition)

$X$ is said to be an Exponential r.v. with $\lambda :X\sim Exp(\lambda )$ if:
$f(x)=\{\begin{array}{ll}\lambda e^{-\lambda x}& x\ge 0\\ 0& x<0\end{array}$

Expectation and Variance

• $E(x)=\frac{1}{\lambda }=\mu$
• $Var(X)=\frac{1}{{\lambda }^2}$

CDF: $F(x)=\{\begin{array}{ll}1-e^{-\lambda x}& x\ge 0\\ 0& x<0\end{array}$

Theorem:

1. Let $N_t$ be the number of arrivals in a time interval of length $t$ of a Poisson process with intensity $\lambda$.
2. Let $T_1$ be the waiting time for the first arrival.

Then, $T_1\sim Exp(\lambda )$