The random variable takes the values 1,2,3 , , m . If P(X=n)= 1 m to each n , then the variance of X is

Step-by-step solution

aligned & var (X)= _ i=2 ^ P_i (X_i- X )^2 & X = 1+2+ +m m = m(m+1) 2 m = m+1 2 & var (X)= P(X=1)+P(X=2)+ +P(X=m) & = 1 m (1- m+1 2 )^2+ 1 m (2- m+1 2 )^2+ + 1 m . & = 1 m (1^2+ (m+1)^2 4 -2 1 ( m+1 2 ) ) . & + (2^2+ (m+1)^2 4 -2 2 m+1 2 )+ & . + (m^2+ (m+1)^2 4 -m 2 (m+1) 2 ) & = 1 m (1^2+2^2+ +m^2 )+ (m+1)^2 4 (1+1+ . & +m times) -(m+1)(1+2+3+ +m) & .= 1 12 & = 1 12 m(m+1)(2 m+1) 6 + m(m+1)^2 4 . & = (m+1)[2(2 m+1)+3(m+1)-6 m+1] 2 aligned