Answer to Question #86833 in Statistics and Probability for Anand

Question

Given that the regression equations of Y on X and of X on Y are respectively Y = X
and 4X − Y = 3, and the second moment of X about the origin is 2, find
i) the means of X and Y;
ii) the correlation coefficient between X and Y;
iii) the standard deviation of Y

Answer 1

(i) Since two regression lines always intersect at a point representing mean values

$$(\bar{x}, \bar{y})$$ $$Y=X$$ $$4X-Y=3$$

Then

$$\bar{x}=1, \bar{y}=1$$

(ii) To find the given regression equations in such a way that the coefficient of dependent variable is less than one at least in one equation. So

$$4X-Y=3\implies4X=3+Y \implies X={3 \over 4}+{1 \over 4}Y$$

That is

$$b_{xy}={1 \over 4}=0.25$$

$$Y=X$$

That is

$$b_{yx}=1$$

Hence coefficient of correlation r between x and y is given by:

$$r=\sqrt{b_{xy}*b_{yx}}=\sqrt{0.25*1}=0.5$$

(iii) To determine the standard deviation of y , consider the formula:

$$\sigma_y={{b_{yx}*\sigma_x} \over r}$$

$${\sigma_x}^2=E[X^2]-(E[X])^2=2-(1)^2=1$$

$$\sigma_x=1$$

$$\sigma_y={1*1 \over 0.5}=2$$