# Poisson Distribution

• Feb 21st 2010, 04:20 AM
mehn
Poisson Distribution
Let X be the poisson random variable with parameter lambda. Obtain E(X^3) and E(X^4).

Please explain to me for the E(X^3) and i'll do the other one by myself.
$M_X(t)=E[e^{tX}]=\sum_{k=0}^\infty e^{tk}\cdot\frac{e^{-\lambda}\lambda^k}{k!}=e^{-\lambda}\sum_{k=0}^\infty \frac{(e^t\lambda)^k}{k!}=e^{-\lambda}e^{\lambda e^t}=e^{\lambda(e^t-1)}$
And use this property : $E[X^n]=\left. M_X^{(n)}(t)\right|_{t=0}$ (n-th derivative at t=0)