Poisson Distribution

**mehn** · February 21st 2010, 03:20 AM

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.
Thank you loads

**Moo** · February 21st 2010, 03:33 AM

Hello,

You can compute the moment generating function :

$M_X(t)=E[e^{tX}]=\sum_{k=0}^\infty e^{tk}\cdot\frac{e^{-\lambda}\lambda^k}{k!}=e^{-\lambda}\sum_{k=0}^\infty \frac{(e^t\lambda)^k}{k!}=e^{-\lambda}e^{\lambda e^t}=e^{\lambda(e^t-1)}$

And use this property : $E[X^n]=\left. M_X^{(n)}(t)\right|_{t=0}$ (n-th derivative at t=0)

Thread: Poisson Distribution

LinkBack

Thread Tools

Display