Exercise Solution 4.22

To calculate E(tX), we start with [4.55]:


We assume the unconditional expectation exists, in which case E(tkX) = E(tX) for all k. Taking the unconditional expectation of both sides of [4.55], we obtain





We have obtained an expression for E(tX) in terms of itself. Rearranging, we conclude


To calculate t–1E(tX), we again start with [4.55]:


Taking the conditional expectation of both sides, we obtain