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 \lambda. Further assume that \lambda varies from location to location and has a gamma distribution w. parameters \alpha and \beta, where \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 E(Y)

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

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

since this is a constant we receive

= E[\alpha\beta] = \alpha\beta

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

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


E[V(Y\mid \lambda)] =
<br />
E[E(Y^2\mid \lambda)] - E{[E(Y\mid \lambda)^2}]
<br />
\alpha\beta^2 + \alpha^2\beta^2 - \alpha\beta^2 =

and 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.