Question

If $x_1,x_2,....,x_n$ are n values of a variable X and $y_1,y_2,....y_n$ are n values of variable Y such that $y_i=a{x}_i+b,i=1,....,n.$ then write Var(Y) interms of Var (X).

Answer 1

$\overline{y}=\frac{1}{n}\{\sum y_i\}=\frac{1}{n}\{\sum a{x}_i+b\}=a\overline{x}+b$\

$\therefore y_i-\overline{y}=a{x}_i+b-a\overline{x}-b=a(x_1-\overline{x})$

$var\left(Y\right)=\frac{1}{n}\{\sum (y_i-\overline{y}{)}^2\}=\frac{1}{n}\{\sum a^2(x_i-\overline{x}{)}^2\}=a^2\left[\frac{1}{n}\{\sum (x_i-\overline{x}{)}^2\}\right]=a^2\left[var\right(x\left)\right]$