Expectation

Expectation

Expected Value

Let $X$ be a random variable with a probability distribution $f(x).$ The mean or expected value of $X$ is

\[\mu = E(X) = \sum_{x}{x f(x)}\]

if $X$ is discrete, and

\[\mu = E(X) = \int_{-\infty}^{\infty}{x f(x) dx}\]

if $X$ is continuous.

The expected value is the “average value” we expect the random variable to take in the long run.

Note that in the discrete case, the expected value is the dot product of the vector of $x$ values and corresponding vector of $f(x)$ values.

Variance

Let $X$ be a random variable with probability distrbution $f(x)$ and mean $\mu.$ The variance of $X$ is

\[\sigma^2 = E[(X - \mu)^2] \sum_{x}{(x - \mu)^2 f(x)}\]

if $X$ is discrete, and

\[\sigma^2 = E[(X - \mu)^2] \int_{-\infty}^{\infty}{(x - \mu)^2 f(x) dx}\]

if $X$ is continuous.

The positive square root of the variance, $\sigma,$ is called the standard deviation of $X.$

Variance and stanard deviation tell us about how spread out the values of $X$ are around its mean.

An alternative and equivalent formula for variance is

\[\sigma^2 = E[X^2] - E[X]^2.\]