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(tkX) = 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]

 

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