Originally Posted by

**Volga** This is a good example of what I am trying to master, therefore, I'd like to try to summarise the approach used here. We are given a conditional distribution for Y on Lambda and told the (marginal?) distribution for Lambda. We are asked to find the (marginal?) distribution of Y.

1. First, if Y is conditioned on Lambda distributed according to some distribution (here Poisson), that means that Y is the random variable in the Poisson formula, as here:

2. Next, we find the joint density according to the standard formula

$\displaystyle f_{Y,\Lambda}(y,\lambda)=f_{Y|\Lambda}(y,\lambda)f _{\Lambda}}(\lambda)$

3. And finally to find $\displaystyle f_Y(y)$ we integrate (or summate) the joint density over the range of Lambda. Whether this is integration or summation will be determined by the type of Lambda variable, right? Here Lambda is exponentially distributed, therefore, continous, therefore, we integrate.

Does it make sense?