Expectation Value

For one discrete variable,

$$\left\langle{f(x)}\right\rangle{} = \sum_x f(x)P(x).$$ (1)

For one continuous variable,

$$\left\langle{f(x)}\right\rangle{} = \int f(x)P(x)\,dx.$$ (2)

The expectation value satisfies

$$\left\langle{ax+by\rangle = a\langle x\rangle +b\langle y}\right\rangle{}$$ (3)

$$\left\langle{a}\right\rangle{} = a$$ (4)

$$\left\langle{\sum x}\right\rangle{} =\sum \left\langle{x}\right\rangle{}.$$ (5)

For multiple discrete variables

$$\left\langle{f(x_1,\ldots,x_n)}\right\rangle{} = \sum_{x_1, \ldots, x_n} f(x_1,\ldots ,x_n)P(x_1,\ldots ,x_n).$$ (6)

For multiple continuous variables

$$\left\langle{f(x_1,\ldots,x_n)}\right\rangle{} = \int f(x_1, \ldots, x_n)P(x_1,\ldots ,x_n)\,dx_1\cdots\,dx_n.$$ (7)

The (multiple) expectation value satisfies

$$\left\langle{(x-\mu_x)(y-\mu_y)}\right\rangle{} = \left\langle{xy-\mu_xy-\mu_yx+\mu_x\mu_y}\right\rangle{}$$

$$= \left\langle{xy}\right\rangle{}-\mu_x\mu_y-\mu_y\mu_x+\mu_x\mu_y$$

$$= \left\langle{xy}\right\rangle{}-\left\langle{x}\right\rangle{}\left\langle{y}\right\rangle{},$$ (8)

where $\mu_i$ is the Mean for the variable $i$.

See also Mean