Law of the Unconscious Statistician (LOTUS)

Idea

Find $\mathrm{E}[g(X)]$ directly using the distribution of $X$ without first having to find the distribution of $g(X)$

Theorem

If $X$ is a discrete r.v. and $g$ is a function from $\mathbb{R}$ to $\mathbb{R}$, then

$$\mathrm{E}[g(X)]=\sum _{x}g(x)P(X=x)$$

where the sum is taken over all possible values of $X$.

This means that we can get the expected value of $g(X)$ knowing only $P(X=x)$, the PMF of $X$; we don't need to know the PMF of $g(X)$.

Proof

Recall the proof in expectation, we have

$$\begin{array}{rl}\mathrm{E}[g(X)]& =\sum _{s}g(X(s))P(\left\{s\right\})\\ & =\sum _s\sum _{s:X(s)=x}g(X(s))P(\left\{s\right\})\\ & =\sum _{x}g(x)\sum _{s:X(s)=x}P(\left\{s\right\})\\ & =\sum _{x}g(x)P(X=x)\end{array}$$