Equivalence of Definitions of Expectation

Theorem

The following definitions of the concept of Expectation are equivalent:

Definition $1$

The expectation of a random variable is the arithmetic mean of its values.

Definition $2$

The expectation of a random variable $X$ is the first moment about the origin of $X$.

Proof

Recall the definition of moment:

Let $X$ be a random variable on some probability space.

Let $a$ be a real number.

Then the $n$th moment of $X$ about $a$, usually denoted $\map {\mu_n} a$, is defined as:

$\map {\mu_n} a = \expect {\paren {X - a}^n}$

where $\expect X$ denotes the expectation of $X$.

Hence the first moment about the origin:

$\map \mu 0 = \expect X$

which is precisely the arithmetic mean.

$\blacksquare$