Karnataka Board PUCPUC Science 2nd PUC Class 12

Variance of a Random Variable




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)`

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