Having some trouble with a hierarchical model. Can someone please check my first answer and help me to get on with the b-assignment?

Assume that Y denotes the number of bacteria per cubic centimeter in a particular liquid and that Y has a Poisson distribution with parameter $\displaystyle \lambda$. Further assume that $\displaystyle \lambda$ varies from location to location and has a gamma distribution w. parameters $\displaystyle \alpha$ and $\displaystyle \beta$, where $\displaystyle \alpha$ is a positive integer. If we randomly select a location, what is the:

a) Expected number of bacteria per cubic centimeter?

That would mean that we're looking for $\displaystyle E(Y)$

$\displaystyle E(Y) = E[E(Y\mid \lambda)] $

since Y given \lambda means that Y also follows the Gamma distribution, we get
$\displaystyle = E[\alpha\beta]$

since this is a constant we receive

$\displaystyle = E[\alpha\beta] = \alpha\beta$

b) standard deviation of the number of bacteria per cubic centimeter?

$\displaystyle V(Y) = E[V(Y\mid \lambda)] + V[E(Y\mid \lambda)]$


$\displaystyle E[V(Y\mid \lambda)] = $
E[E(Y^2\mid \lambda)] - E{[E(Y\mid \lambda)^2}]$
\alpha\beta^2 + \alpha^2\beta^2 - \alpha\beta^2$ =
$\displaystyle \alpha^2\beta^2$

and $\displaystyle V[E(Y\mid\lambda)] = E\left\{[E(Y\mid\lambda)]^2 \right\} - \left\{E[E(Y\mid\lambda)] \right\}^2$

but how do I calculate this? I get confused by all those parenthesis.

Thanks for your help.