Let Z be a standard normal random variable under a probability measure P. Set X = exp(θZ - θ2/2). Define a new probability measure P1(A) = E(X 1A), where the expected value is taken under P and 1A denotes the indicator function.

Now, let Y = Z - θ and find the moment generating function of Y under P1 and deduce the distribution of Y under P1.

Any help is appreciated.


closed as off-topic by Did, Shailesh, Leucippus, GNUSupporter 8964民主女神 地下教會, Claude Leibovici Apr 8 '18 at 7:13

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  • 1
    $\begingroup$ Who is $A$? $\;$ $\endgroup$ – Math1000 Apr 7 '18 at 22:02
  • $\begingroup$ @Math1000 what does your question even mean? $\endgroup$ – Jason Apr 7 '18 at 22:17
  • $\begingroup$ @Jason Do you know who $A$ is? $\endgroup$ – Math1000 Apr 7 '18 at 22:19
  • 1
    $\begingroup$ A would be a set in this case. $\endgroup$ – TedMosby Apr 7 '18 at 22:36

Hint: if $E$ and $E_1$ denote expectation with respect to $P$ and $P_1$ respectively, then $E_1[W]=E[WX]$ for any random variable $W$. Apply this with $W=\exp(\lambda Y)$ to calculate the MGF of $Y$.


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