SOA 130

The solution given is pretty straightforward and obviously the easiest way to solve this problem, but I recognized the MGF as an exponential random variable with mean = 2 which made me want to use one of the following 2 methods to solve it:



1. Use the properties of expectation and the natural log function to solve for E[Y].



X~exp(lambda=1/2)

E[X] = 2

E[100(0.5)^X] = c



Im pretty sure this method won't work because lnE[X] does not necessarily equal E[lnX] for non-linear functions, according to Jensen's inequality.



2. Method of Transformations



Since the distribution of X is known, I decided to use the method of transformations. I let Y = 100(0.5)^x and solved for the pdf of y, and then I tried to integrate y*f(y) to get E[Y], but I could not integrate it successfully. I was wondering if someone else could give the integration a try.

SOA 130