Name: Scatter Graphs
Uploaded: 2019-07-19
Duration: 6 min 43 s
Description: Scatter plots are a powerful method of visualising relations between sets of numeric values. In this video, we will learn to construct a scatter plot.

Question

Prove that though covariance is independent of the choice of origin it depends upon the scale. If $u = a x + b, v = c y + d,$ show that $cov (u, v) = a . c . cov (x, y)$ .

Accepted Answer

Let $u = x + A; v = c y + B$

$∴ u = \bar{x} + A; \bar{v} = \bar{y} + B \Rightarrow u - u = x - \bar{x}; v - \bar{v} = y - \bar{y}$

We know that,

$∴ Cov (u, v) = \frac{\sum (u - \bar{u}) (v - \bar{v})}{n} = \frac{\sum (x - \bar{x}) (y - \bar{y})}{n} = Cov (x, y)$

$\Rightarrow Cov (x + A, y + B) = Cov (x, y)$

Thus, covariance is independent of choice of origin.

Let $u = a x$ and $v = b y$ and $u = a \bar{x}$ and $v = b \bar{y}$

$∴ u - \bar{u} = a (x - \bar{x}); v - \bar{v} = b (y - \bar{y})$

$∴ Cov (u, v) = \frac{\sum (u - \bar{u}) (v - \bar{v})}{n} = \frac{\sum a b (x - \bar{x}) (y - \bar{y})}{n}$

$= \frac{a b \sum (x - \bar{x}) (y - \bar{y})}{n} = a . b . cov (x, y)$

Thus, clearly covariance depends upon the scale

if $u = a x + b; v = c y + d$

$∴ \bar{u} = a \bar{x} + b; \bar{v} = c \bar{y} + d$

$\Rightarrow u - \bar{u} = a (x - \bar{x}); v - \bar{v} = c (y - \bar{y})$

Thus, $cov (u, v) = \frac{\sum (u - \bar{u}) (v - \bar{v})}{n} = \frac{\sum a c (x - \bar{x}) (y - \bar{y})}{n} = a . c . cov (x, y)$

Hence, proved

$x$	$1$	$2$	$3$	$4$	$5$
$y$	$2$	$5$	$3$	$8$	$7$

Sales (in lakh of ₹)	$18$	$20$	$27$	$20$	$21$	$29$
Expenditure (in lakh of ₹)	$23$	$27$	$28$	$28$	$29$	$30$

Prove that though covariance is independent of the choice of origin it depends upon the scale. If $u = a x + b, v = c y + d,$ show that $cov (u, v) = a . c . cov (x, y)$ .

Important Questions on Correlation Analysis

Learn and Prepare for any exam you want !

Prove that though covariance is independent of the choice of origin it depends upon the scale. If u=ax+b, v=cy+d, show that cov u, v=a.c.cov x, y.

Important Questions on Correlation Analysis

Learn and Prepare for any exam you want !

Prove that though covariance is independent of the choice of origin it depends upon the scale. If $u = a x + b, v = c y + d,$ show that $cov (u, v) = a . c . cov (x, y)$ .