Are you trying to see if the second derivative is correct?
I think I'm doing this wrong, but I can't see where. Y is a random variable.
m'(t) = Expected Value (50 * e^(t*50) + 0.01 * Y * e^(t*0.01*Y))
m''(t) = Expected Value (50^2 * e^(t*50) + (0.01 * Y)^2 * e^(t*0.01*Y))