Given that is the mean and σ² is the variance of n observations x₁, x₂ … x_n. Prove that the mean and variance of the observations ax₁, ax₂, ax₃ …ax_nare and a² σ², respectively (a ≠ 0).

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Solution

The given n observations are x₁, x₂ … x_n.

Mean =

Variance = σ²

If each observation is multiplied by a and the new observations are y_i, then

Therefore, mean of the observations, ax₁, ax₂ … ax_n, is .

Substituting the values of x_iand in (1), we obtain

Thus, the variance of the observations, ax₁, ax₂ … ax_n, is a² σ².

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If the mean and variance of the observations $x_{1}, x_{2}, x_{3}, \dots, x_{n}$ are $¯ x$ and $σ^{2}$ respectively and a be a nonzero real number, then show that the mean and variance of $a x_{1}, a x_{2}, a x_{3}, \dots . a x_{n}$ are $¯ a x$ and $a^{2} σ^{2}$ respectively.

Q. 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).

Let $x_{1}, x_{2}, \dots, x_{n}$ be n observations, and let $¯ x$ be their arithmetic mean and $σ^{2}$ be the variance.
Statement – 1: Variance of $2 x_{1}, 2 x_{2}, \dots, 2 x_{n} is 4 σ^{2}$
Statement – 2: Arithmetic mean $2 x_{1}, 2 x_{2}, \dots, 2 x_{n} is 4 ¯ x$