Given that `barx` is the mean and `sigma^(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 `abarx` and `a^(2)sigma^(2)` , respectively, `(a!=0)`

Explanation

To prove that the mean and variance of the observations \( aX_1, aX_2, \ldots, aX_n \) are \( a\bar{x} \) and \( a^2\sigma^2 \), respectively, we will follow these steps:

### Step 1: Define the Mean
The mean \( \bar{x} \) of the observations \( X_1, X_2, \ldots, X_n \) is given by:
\[ \bar{x} = \frac{X_1 + X_2 + \ldots + X_n}{n} \]

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