Exercise Solution 4.22
To calculate E(tX), we start with [4.55]:
[4.55]
We assume the unconditional expectation exists, in which case E(t–kX) = E(tX) for all k. Taking the unconditional expectation of both sides of [4.55], we obtain
[s1]
[s2]
[s3]
[s4]
We have obtained an expression for E(tX) in terms of itself. Rearranging, we conclude
[s5]
To calculate t–1E(tX), we again start with [4.55]:
[4.55]
Taking the conditional expectation of both sides, we obtain
[s6]
[s7]
[s8]
[s9]