# Moment Generating Function under a new probability measure [closed]

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 LeiboviciApr 8 '18 at 7:13

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Shailesh, Leucippus, GNUSupporter 8964民主女神 地下教會, Claude Leibovici
If this question can be reworded to fit the rules in the help center, please edit the question.

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