Moment Generating Function under a new probability measure

Question

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 1_A), where the expected value is taken under P and 1_A 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.

Answer 1

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 $Y$.