Question

Given that is the mean and σ 2 is the variance of n observations x 1 , x 2 … x n . Prove that the mean and variance of the observations ax 1 , ax 2 , ax 3 … ax n are and a 2 σ 2 , respectively ( a ≠ 0).

Answer 1

Given, mean and variance of n observations x 1 ,  x 2 ........... x n are x ¯ and σ 2 respectively.

Multiply each observation by a and consider new observations as y i , then

y i =a x i x i = 1 a y i

So, new mean of the observations a x 1 , a x 2 ...........a x n is,

y ¯ = 1 n ∑ i=1 n y i = 1 n ∑ i=1 n a x i = a n ∑ i=1 n x i =a x ¯

Substitute the values of x i and x ¯ , and determine the variance,

Variance( σ 2 )= 1 n ∑ i=1 n ( x i − x ¯ ) 2 = 1 n ∑ i=1 n ( 1 a y i − 1 a y ¯ ) 2 a 2 σ 2 = 1 n ∑ i=1 n ( y i − y ¯ ) 2

Hence, the variance of observations a x 1 ,a x 2 ,.........................a x n ,is a 2 σ 2 .