Conditional expectation and Iterated expectation problem

Question

I have $X_i|b_i \sim Poisson(\lambda_i)$, and $log(\lambda_i)=x_i^t\beta+\varepsilon_i$ further $\varepsilon\sim N(0,\sigma^2)$

then, how to find $E(X_i)$?

I try using the iterated expectation $E(X_i)=E[E(X_i|b_i)]$, so $E(X_i)=E(\lambda_i)$ but how to use $log(\lambda)$ to find the solution?

thanks.

Answer 1

Your iterated expectations strategy looks good.

We have $\lambda_i = Ce^{\epsilon_i}$ where $C=e^{x_i^t\beta}$ so can take the expected value $$E(\lambda_i) = CE(e^{\epsilon_i})=C\int_{-\infty}^\infty e^t \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{t^2}{2\sigma^2}}dt $$ since $\epsilon_i\sim N(0,\sigma^2).$ The integral is a Gaussian integral and can be done by completing the square.