Use conditional expectation to compute the moment generating function M_z(s) of the random variable Z=XT.
X ~ R(0,10)
T ~ exp(0.1)
The Attempt at a Solution
M_z(s) = E(exp(sZ))
The only thing I can think of doing is using double expectation.
So M_z(s) = E[E(exp(sXT|X)] = E[E(exp(sXT|T)]...(1)
But here I'm stuck. Both X and T are continuous random variables so I can't just plug in the values of X and T one by one. Writing (1) as an integral of an expectation doesn't help either because I don't have an expression for the integrand and so cannot evaluate the integral. This question is really doing me in and any help to relieve my great pain would be much appreciated.