law of the unconscious statistician

LOTUS
#statistics #probability

Theorem

Suppose a random variable $X$ .

If a probability distribution of $X$ is discrete, and one knows its probability mass function $p_X$ , then expected value of $g(X)$ is

\mathbb{E}[g(X)] = \sum_x g(x) p_X(x)

If a probability distribution of $X$ is continuous, and one knows its probability density function $f_X(x)$ , then expected value of $g(X)$ is

\mathbb{E}[g(X)] = \int _{-\infty }^{\infty }g(x)f_{X}(x)\,\mathrm {d} x

References

https://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician