Yes, you use law of iterated expectations. Then you use that the conditional (on N) distribution of the sum is Poisson (sum to n of 2i-1). So the mean of the cond. distribution is sum to N of 2i-1. Then you just take the expectation wrt N of that!
I am in desperate need of help!
Suppose that X1,X2, . . . ,X100 are independent random variable, that Xi is
Poisson distributed with mean 2i − 1, and that N is another independent
random variable with N Bin(100, p). Let X = SUM(Xi) form i=1 to N .
How do I find the mean of X? Do I use the tower law? I don't know where to go from there though.
Yes, you use law of iterated expectations. Then you use that the conditional (on N) distribution of the sum is Poisson (sum to n of 2i-1). So the mean of the cond. distribution is sum to N of 2i-1. Then you just take the expectation wrt N of that!