Given that ` bar x`is the mean and `sigma^2`is the variance of n observations `x_1x_2`, ..., `x_n`. Prove that the mean and variance of the observations `a x_1,a x_2`, `a x_3,dotdotdot,a x_n`are `a bar x`and `a^2sigma^2`,

Explanation

To prove that the mean and variance of the observations \( a x_1, a x_2, \ldots, a x_n \) are \( a \bar{x} \) and \( a^2 \sigma^2 \) respectively, we will follow these steps:

### Step 1: Understand the given information
We are given \( \bar{x} \) as the mean and \( \sigma^2 \) as the variance of the observations \( x_1, x_2, \ldots, x_n \).

- The mean \( \bar{x} \) is defined as:
\[
\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
...