Question.

Let Y be a random variable that has a Poisson distribution with parameter $\displaystyle \lambda$. Without using generating functions:

(a) show that E(X) = $\displaystyle \lambda$

(b) find E(X^3).

Answer

(a)

$\displaystyle E(X)=\Sigma_xxf_X(x)=\Sigma_{x=0}^{\infty}\frac{\l ambda^xe^{-\lambda}x}{x!}={\lambda}e^{-\lambda}\Sigma_{x=0}^{\infty}\frac{\lambda^{x-1}}{(x-1)!}={\lambda}e^{-\lambda}e^{\lambda}={\lambda}e^0=\lambda$

(I used exp expansion for summation)

(b) and now I am stuck with the powers of x in my summation expression...

$\displaystyle E(X)=\Sigma_xx^3f_X(x)=\Sigma_{x=0}^{\infty}\frac{ \lambda^xe^{-\lambda}x^3}{x!}={\lambda}e^{-\lambda}\Sigma_{x=0}^{\infty}\frac{\lambda^{x-1}x^2}{(x-1)!}$

I tried to substitute x-1=j:

$\displaystyle ={\lambda}e^{-\lambda}\Sigma_{j=1}^{\infty}\frac{\lambda^j(j+1)^ 2}{j!}=$

and then possibly expand (j+1)^2 into {j^2+2j+1} but then I have to deal with j^2 again (as with x^2 before), so it does not seem to help.

Are there any methods to deal with these kinds of summation calculations? I have looked into many books but they all show calculations of n-th moments straight from the moment generating functions, and my examinor seem to prefer his students to calculate from 'first principles' - this is a sample exam question. Unfortunately, there is no answer to this question in the study guide.