Question

Evaluate E{X(X-1)} for a poisson random variable with mean of Lambda , and hence, derive its variance

Answer 1

Variance is given by
\[\begin{align*}V(X)&=E(X^2)-\bigg(E(X)\bigg)^2\\
&=E(X^2)-\lambda^2\end{align*}\]
Rewriting the given expression, you have
\[\begin{align*}E\bigg(X(X-1)\bigg)&=E(X^2-X)\\
&=E(X^2)-E(X)\\
&=E(X^2)-\lambda
\end{align*}\]
You should be able to use the Poisson PDF from here to figure out the variance.