Question

Prove that coefficient of correlation lies between $-1$ and $1$.

Answer 1

$p=\frac{\sum (x.\overline{x})(y-\overline{y})}{\sqrt{\sum {(x.\overline{x})}^2\sum {(y-\overline{y})}^2}}$
$=\frac{\sum XY}{\sqrt{\sum X^2\sum Y^2}}$
Where $(X=(x.\overline{x});Y=(y-\overline{y})$
$\therefore$ by Swchwarz's inequality
$\left(S{X}^2{Y}^2\right)\le \sum X^2\sum Y^2$
$\frac{{(\sum XY)}^2}{\sum X^2\sum Y^2}\le 1$
$p^2\le 1$
$\Rightarrow$ $-1\le p\le 1$
Hence value of correlation coefficient lies between $-1$ and $1$