Another hierarchical model

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 . Further assume that varies from location to location and has a gamma distribution w. parameters and , where 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

since Y given \lambda means that Y also follows the Gamma distribution, we get

since this is a constant we receive

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

where

=

and

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

Thanks for your help.