#### definition

Let X be a random variable whose possible values `x_1, x_2,...,x_n` occur with probabilities `p(x_1), p(x_2),..., p(x_n)` respectively.

Let µ = E (X) be the mean of X. The variance of X, denoted by Var (X) or `σ_x^2` is defined as

`σ_x^2 = Var (X) = sum_(i=1)^n (x_i - mu )^2 p (x_i)`

or equivalently `σ_x^2` = `E(X - mu)^ 2 `

The non-negative number

`σ_x = sqrt (Var (X)) = sqrt (sum_(i=1)^n (x_i - mu)^2 p (x_i))`

is called the standard deviation of the random variable X. **Another formula to find the variance of a random variable.** We know that,

Var (X) = `sum_(i=1)^n (x_i - mu )^2 p(x_i)`

`= sum_(i=1) ^n (x_i^2 + mu ^2 - 2 mu x_i ) p(x_i)`

`= sum_(i=1)^n x_i^2 p(x_i) + sum_(i =1)^n mu^2 p(x_i) - sum_(i=1)^n 2 mu x_i p(x_i)`

`= sum_(i =1)^n x_i^2 p (x_i) + mu ^2 sum_(i=1)^n p(x_i) - 2 mu sum_(i=1)^n x_i p(x_i)`

`= sum_(i=1)^n x_i^2 p (x_i) + mu ^2 - 2 mu^2 [ "since" sum_(i = 1)^n p(x_i) = 1 and mu = sum_(i=1)^n x_i p(x_i)]`

`= sum_(i=1)^n x_i^2 p(x_i) - mu^2`

or Var (X) = `sum_(i=1)^n x_i^2 p(x_i) - (sum _(i=1)^n x_i p(x_i))^2`

`or Var (X) = E(x^2) - [E(X)] ^2 , "where" E(X^2) = sum_(i=1)^n x_i^2 p(x_i)`