D2 - variance

variance

the variance of RV $X$ is the expected squared difference of $X$ from its mean:

Var (X) = E [(X - E [X])^{2}]

the same measurement taken in different scales will have different values of variance
eg: $X = {3, 2, 1}$ and $Y = {30, 20, 10}$ , where each event has a probability of $1 / 3$
$Var (X) = 2 / 3$ and $Var (Y) = 200 / 3$

standard deviation

it is the square root of variance:

σ_{X} = std (X) = \sqrt{Var (X)}

again, $σ_{X} = \sqrt{2 / 3}$ while $σ_{Y} = \sqrt{200 / 3}$

squared coefficient of variation

the squared coefficient of variation of RV $X$ is:

C_{X}^{2} = \frac{Var (X)}{E [X^{2}]}

now, $C_{X}^{2} = C_{Y}^{2} = 1 / 6$

equivalent definition lemma

$Var (X) = E [X^{2}] - (E [X])^{2}$

linearity of variance

let $X$ and $Y$ be RVs such that $X ⊥ Y$

Var (X + Y) = Var (X) + Var(Y)

in general:

Var (X_{1} + X_{2} + \dots + X_{n)} = Var (X_{1}) + Var (X_{2}) + \dots + Var (X_{n})