Given that is the mean and is the variance of n observations x₁, x₂, ...,x_n.

Prove that the mean and variance of the observations ax₁, ax₂, ax₃, ...., ax_n are and , respectively,

The given n observations in the question are x₁, x₂,…..x_n

Also, mean =

And, variance = σ²

We know that,

According to the condition given in the question, if each of the observation is being multiplied by a and the new observation are y_i the, we have:

y_i = ax_i

Hence, mean of the observations ax₁, ax₂,…..ax_n is a

Now, by substituting the values of x_i and in (i), we get:

Hence, the variance of the given observations ax₁, ax₂,….ax_n is a²σ²