# How to show $P_{\hat X}=P_{X}$, where $P_{X}$ is a known distribution?

Let $X$ be a random variable on the probability space $(\Omega,\mathcal B,P)$, with distribution $P_{X}$. Consider the random variable $\hat X$ on the probability space $(\mathbb R,\mathcal B_{\mathbb R},P_{X})$,defined by $\hat X(x)=x$ . Then $P_{\hat X}=P_{X}$.

$\mathcal B$ is $\sigma$-algebra.

• This has already been asked and answered here. – Stefan Hansen Dec 13 '13 at 11:02