A insurance company offers ship owners insurance for broken down ships. For the calculation assume that the reported broken down ships follow a poission process. On average a breakdown happens on $1$ day out of $1000$ days and the damage is $\text{Binomial($2\cdot 10^7,0.5$)}$ distributed and is independent of the damage of other ships and independent of the poisson process. Calculate the expected damage in one year ($365$ days).
I am having trouble "merging" the poisson process with the binomial damage claim. I basically have two random variables $$X:=\text{number of broken down ships} \sim Poi(\frac{1}{1000}) \\ Y:= \text{damage of broken down ship} \sim Bin(2\cdot 10^7,0.5)$$
How can I put these together to find the expected damage?
If there are a random number $X$ of damaged ships, and each one has damage $Y_i$, then the total damage is
$$\mathbf E \left[ \sum_{i=1}^X Y_i \right].$$
Wald's equation allows you to calculate the expectation of a random sum of random variables, under the condition that the terms of the sum are independent and identically distributed.
$$\mathbf E \left[ \sum_{i=1}^X Y_i \right] = \mathbf E[X] \mathbf E[Y].$$
In your case this becomes
$$\mathbf E \left[ \sum_{i=1}^X Y_i \right] = \left(\frac{365}{1000}\right) \times \left( \frac12 \times 2 \cdot 10^7\right) = 3.65 \times 10^6$$