I am currently learning Monte Carlo integration, I am completely lost on how to calculate the expected value of the following, before I simulate it using Monte Carlo.
$R0 = px + (1-p)exp(\epsilon +x)$
where x has a $Beta(3,1)$ distribution where B(3,1) = 1/3.
I can calculate $P(X=x) = 3x^2$ using the pmf of a beta distribution, but i get lost when it comes to using this in the integration alongside $\epsilon$.
I know the output is know that the output is $E[R0]=3(p/4+(1−p)(exp(ϵ+1)−2exp(ϵ)))$ but I have no idea what steps were taken to get there.
Am more than happy to take general integration guidance / sources to learn from - havnt been able to find this specific problem anywhere and am trying to work through each step to learn.
Any help would be much appreciated!