Change of Variables for Expectations

Let $X$ be a random variable on a probability space $(\Omega,\mathcal{F},P)$ with distribution $P_X$, let $\mathbb{R}$ denote the real numbers, and let $\varphi:\mathbb{R}\to\mathbb{R}$ be measurable with respect to the Borel $\sigma$-algebra on both sides. Then $\varphi\circ X$ is a random variable (preimages compose), and, with the expectation notation $\mathbb{E}$:

1. if $\varphi\ge 0$, then, with integrals as in Lebesgue Integral of a Nonnegative Measurable Function,

2. in general, $\varphi\circ X$ is integrable with respect to $P$ if and only if $\varphi$ is integrable with respect to $P_X$, and in that case the displayed identity holds in $\mathbb{R}$.

In particular, expectations, moments, and variances of random variables depend only on their distributions, and identically distributed random variables share them.