Answer to Question #86833 in Statistics and Probability for Anand

Question #86833
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
1
Expert's answer
2019-03-25T15:32:41-0400

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


(xˉ,yˉ)(\bar{x}, \bar{y})Y=XY=X4XY=34X-Y=3

Then


xˉ=1,yˉ=1\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


4XY=3    4X=3+Y    X=34+14Y4X-Y=3\implies4X=3+Y \implies X={3 \over 4}+{1 \over 4}Y

That is


bxy=14=0.25b_{xy}={1 \over 4}=0.25

Y=XY=X

That is


byx=1b_{yx}=1

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


r=bxybyx=0.251=0.5r=\sqrt{b_{xy}*b_{yx}}=\sqrt{0.25*1}=0.5

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


σy=byxσxr\sigma_y={{b_{yx}*\sigma_x} \over r}

σx2=E[X2](E[X])2=2(1)2=1{\sigma_x}^2=E[X^2]-(E[X])^2=2-(1)^2=1

σx=1\sigma_x=1

σy=110.5=2\sigma_y={1*1 \over 0.5}=2


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