The discrete random variable X can take all possible integer values from 1 to k , each with a probability 1 k , then its variance is

Step-by-step solution

aligned & E ( X )= _ i =1 ^ k x_ i p _ i = 1 k + 2 k + + k k = k +1 2 & E ( X ^2 )= _ i =1 ^ k x_ i ^2 p _ i = 1^2+2^2+3^2+ + k ^2 k & = ( k +1)(2 k +1) 6 & Variance = E ( X ^2 )-[ E ( X )]^2 & = 2 k ^2+3 k +1 6 - k ^2+2 k +1 4 & aligned aligned & = 4 k ^2+6 k +2-3 k ^2-6 k -3 12 & = k ^2-1 12 aligned