Let x_1,x_2,,x_n be n observations, and let x be their arithmetic mean and ^2 be the varia…

Question

Let $x_1,x_2,\ldots,x_n$ be n observations, and let $\bar x$ be their arithmetic mean and $\sigma^2$ be the variance.

Statement-1: Variance of $2x_1,2x_2,\ldots,2x_n$ is $4\sigma^2$

Statement-2: Arithmetic mean of $2x_1,2x_2,\ldots,2x_n$ is $4\bar x$.

Accepted Answer

A.M. of $2x_1,\ldots,2x_n$ is $2\bar x$, not $4\bar x$ --- so Statement-2 is false.

If each observation is multiplied by 2, mean is multiplied by 2 and variance by $2^2=4$: $	ext{Var}(2x_i) = 4\sigma^2$ --- so Statement-1 is true.

Solutions · 1