Question

Let $x_1,x_2,.....x_n$ be values taken by a variable X and $y_1,y_2,.....y_n$ be the values taken by a variables Y such that $y_i=a{x}_i+b,i=1,2,.....,n.$ Then.

• A. $Var\left(Y\right)=a^{2}Var\left(X\right)$
• B. Var (X) = $a^{2}Var\left(Y\right)$
• C. Var(X) = Var (X) + b
• D. none of these

Answer 1

The correct option is A

$Var\left(Y\right)=a^{2}Var\left(X\right)$

$Var\left(X\right)=\frac{{\sum }_{i-1}^n(x_i-\overline{X}{)}^2}{n}$ where mean $\left(\overline{X}\right)=\frac{{\sum }_{i-1}^n{x}_i}{n}$

Var (Y) =$\frac{{\sum }_{i=1}^n(y_i-\overline{Y}{)}^2}{n}$ and $\overline{Y}=\frac{{\sum }_{i-1}^n{y}_i}{n}$

We have,

$y_i=a{x}_i+b$

$\overline{Y}=\frac{{\sum }_{i-1}^n{y}_i}{n}$

$=\frac{{\sum }_{i-1}^{n}a{x}_i+b}{n}=\frac{a{\sum }_{i-1}^n{x}_i}{n}+\frac{nb}{n}=a\overline{X}+b$

Var (Y) = $\frac{{\sum }^{}i-1^n(y_i-\overline{Y}{)}^2}{n}$

$=\frac{{\sum }_{i-1}^n{\{a{x}_i+b-(a\overline{X}+b\left)\right\}}^2}{n}$

$=\frac{{\sum }_{i-1}^n(a{x}_i-a\overline{X}{)}^2}{n}$

$=a^2\frac{{\sum }_{i-1}^n(x_i-\overline{X}{)}^2}{n}$

$=a^{2}Var\left(X\right)$