Poisson random variable
PoissonRandomVariable
2001-10-26 03:32:33
2006-12-16 12:06:15
Definition
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The Poisson discrete probability function with parameter $\lambda>0$ is given by $$f_X(x) = \frac{e^{-\lambda} \lambda^x}{x!},\quad\quad x\in \mathbb{N}.$$
A random variable $X$ with such a density has expectation, variance, moment generating function and characteristic function given by $E[X] = \lambda$, $Var[X] = \lambda$, $M_X(t) = e^{\lambda (e^t - 1)}$, and $\phi_X(t) = e^{\lambda(e^{it}-1)}$, respectively.