Expected Value

The expectation or mean of a random variable summarizes its distribution by a single number — its center of mass.

Discrete Random Variables

The expectation of a discrete random variable is defined as

\[\mathbb{E}[X]=\sum_{x}xp_X(x).\]

Properties of Expectations

• If \(X\geq0\), then \(\mathbb{E}[X]\geq0\).
• If \(a\leq X\leq b\), then \(a\leq\mathbb{E}[X]\leq b\).
• If \(c\) is a constant, then \(\mathbb{E}[c]=c\).
• Expectation of \(g(X)\)

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

Warning

In general, \(\mathbb{E}\!\left[g(X)\right]\neq g\left(\mathbb{E}[X]\right)\).

• Linearity

$\(\mathbb{E}[aX+b]=a\,\mathbb{E}[X]+b\)$

Continuous Random Variables

Instead of taking sums, we use integration to calculate the expectation of a continuous random variable, i.e.,

\[\mathbb{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\,\mathrm{d}x.\]

Properties of the Expectation

• If \(X\geq 0\), then \(\mathbb{E}[X]\geq 0\).
• If \(a\leq X\leq b\), then \(a\leq\mathbb{E}[X]\leq b\).
• Given a function \(g(X)\),

$\(\mathbb{E}[g(X)]=\int_{-\infty}^{\infty}g(x)f_X(x)\,\mathrm{d}x.\)$

• Linearity

$\(\mathbb{E}[aX+b]=a\,\mathbb{E}[X]+b\)$