Math Formulas
Variance
Variance of Continuos Random Variable:
$Var(X)~=~\sigma^2$ $={\large\int}^ \infty_{-\infty} ~(x-\mu)^2f(x)~dx$ $={\large\int} ~x^2f(x)~dx -{\mu}^2 $
where, $\mu=~$ Expected Value
$f$ is the probability density function of $X$.
$E(X)~=\mu~={\large\int}^ \infty_{-\infty}~xf(x)~dx \\$
Standard Variation:
$sd(X) = \sqrt{ var(X)}$
Variance of Discrete Random Variable:
$Var(X)~=~\sigma_x^2~$ $=~\sum\limits_{i=1}^n(x_i~-~\mu)^2 p_i$
where,
$x_i~=~$Outcomes
$p_i~=~$Probabilities
Relation Between Variance and Expectation:
$Var(X)~ =~ E(X^2)~ – ~[E(X)]^2$
$E(X^2)~=~{\large\int}^ \infty_{-\infty} ~x^2f(x)~dx$
$\\ \\$
$E(X)~=~{\large\int}^ \infty_{-\infty} ~xf(x)~dx$
Properties of Variance:
When X and Y are independent random variables:
$Var(X+Y) = Var(X) + Var(Y)~~~~~~$
Non-negative Properties:
$ Var(X) \ge 0 $
If a and b are constants,
$Var(a + b X) = b^2~Var(X)$