Here's how far I have gotten, which isn't very btw.

T= length someone remembers an actuarial statistic

T|Y has an exp distribution with mean 1/Y so f(t|y)=ye^(-ty)

f(y)=4ye^(-2y)

We need to figure out P(T<1/2).

I figure I could find f(t) by multiplying f(t|y) and f(y) to get f(t,y) and then integrating that with respect to y to get f(t). But the integral looks too messy.

Please explain this to me without the use of MGF. The manual I have explains it using MGF which isn't making any sense to me.

Thanks in advance!