Moments

The rth moment of a random variable $X$ ( $μ^{'}$ ) is the expected value of $X^{r}$

\begin{aligned} μ_{r}^{'} = E (X^{r}) = \sum_{x} x^{r} f (x) & (d i s c r e t e) \\ μ_{r}^{'} = E (X^{r}) = \int_{- \infty}^{\infty} x^{r} f (x) d x & (c o n t i n u o u s) \end{aligned}

for $r = 0, 1, 2 \dots$

The rth moment about the mean of a random variable $X$ ( $μ_{r}$ ) is $E [(X - μ)^{r}]$

\begin{aligned} μ_{r} = E [(X - μ)^{r}] = \sum_{x} (x - μ)^{r} \cdot f (x) & (d i s c r e t e) \\ μ_{r} = E [(X - μ)^{r}] = \int_{- \infty}^{\infty} (x - μ)^{r} \cdot f (x) d x & (c o n t i n u o u s) \end{aligned}

The second central moment is the variance of $X$

var (X) = σ^{2} = μ_{2} = E [(X - μ)^{2}] = E (X^{2}) - [E (X)]^{2}

$E (X^{2} - 2 X μ + μ^{2}) = E (X^{2}) - 2 E (X) μ + μ^{2} = E (X^{2}) - [E (X)]^{2}$ as $μ = E [X]$

Properties

var (a X + b) = a^{2} [E (X^{2}) - [E (X)]^{2}] = a^{2} σ^{2} = a^{2} var (X)

v a r (a X + b Y) = a^{2} v a r (X) + 2 a b cov (X, Y) + b^{2} v a r (Y)

If $X$ and $Y$ are independent, the Covariance is 0, and the variance simplifies