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2.6.1 Expectation

Expectation is a measure of the location/mean of the random variable (in the units of the random variable). The expected value of a discrete random variable $R$ is

	$\displaystyle\operatorname{\mathsf{E}}\left[{R}\right]$	$\displaystyle=\sum_{\omega\in\Omega}R(\omega)\operatorname{\mathsf{P}}\left({% \omega}\right)$
		$\displaystyle=\sum_{r=-\infty}^{\infty}\sum_{\omega\in\Omega\colon R(\omega)=r% }R(\omega)\operatorname{\mathsf{P}}\left({\omega}\right)$
		$\displaystyle=\sum_{r=-\infty}^{\infty}r\sum_{\omega\in\Omega\colon R(\omega)=% r}\operatorname{\mathsf{P}}\left({\omega}\right)$
		$\displaystyle=\sum_{r=-\infty}^{\infty}rp_{R}(r).$

Any real-valued function $g(R)$ is also a random variable, $G$ , say, and, by the same line of argument, its expectation is

\displaystyle\operatorname{\mathsf{E}}\left[{g(R)}\right]=\operatorname{% \mathsf{E}}\left[{G}\right]=\sum_{\omega\in\Omega}G(\omega)\operatorname{% \mathsf{P}}\left({\omega}\right)=\sum_{\omega\in\Omega}g(R(\omega))% \operatorname{\mathsf{P}}\left({\omega}\right)=\sum_{r=-\infty}^{\infty}g(r)p_% {R}(r).

This is sometimes referred to as The Law of the Unconscious Statistician.

The expected value of a continuous random variable $X$ is

\displaystyle\operatorname{\mathsf{E}}\left[{X}\right]=\int_{-\infty}^{\infty}% tf_{X}(t)\,\mathrm{d}t.

Similarly for a real-valued function $g(X)$ of a continuous random variable $X$ the expected value is

\displaystyle\operatorname{\mathsf{E}}\left[{g(X)}\right]=\int_{-\infty}^{% \infty}g(t)f_{X}(t)\,\mathrm{d}t.

A proof of the above Law of the Unconscious Statistician for continuous random variables is given in Appendix B. As an example application e.g.:

\displaystyle\operatorname{\mathsf{E}}\left[{\cos(X)}\right]=\int_{-\infty}^{% \infty}\cos(t)f_{X}(t)\,\mathrm{d}t.