n Example 1.6.6, we began with a standard normal random variable $X$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and defined the random variable $Y=X+\theta$, where $\theta$ is a constant. We also defined $Z=e^{-\theta X-\frac{1}{2} \theta^2}$ and used $Z$ as the Radon-Nikodým derivative to construct the probability measure $\widetilde{\mathbb{P}}$ by the formula (1.6.9):
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1 General Probability Theory
$$
\tilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) \text { for all } A \in \mathcal{F} .
$$
Under $\widetilde{\mathbb{P}}$, the random variable $Y$ was shown to be standard normal.
We now have a standard normal random variable $Y$ on the probability space $(\Omega, \mathcal{F}, \widetilde{\mathbb{P}})$, and $X$ is related to $Y$ by $X=Y-\theta$. By what we have just stated, with $X$ replaced by $Y$ and $\theta$ replaced by $-\theta$, we could define $\widehat{Z}=e^{\theta Y-\frac{1}{2} \theta^2}$ and then use $\widehat{Z}$ as a Radon-Nikodým derivative to construct a probability measure $\widehat{\mathbb{P}}$ by the formula
$$
\widehat{\mathbb{P}}(A)=\int_A \widehat{Z}(\omega) d \tilde{\mathbb{P}}(\omega) \text { for all } A \in \mathcal{F},
$$
so that, under $\widehat{\mathbb{P}}$, the random variable $X$ is standard normal. Show that $\widehat{Z}=\frac{1}{Z}$ and $\widehat{\mathbb{P}}=\mathbb{P}$.