# The Two Lines of Regressions Are X + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the Variance of X is 12. Find the Variance of Y and the Coefficient of Correlation. - Mathematics

Sum

The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.

#### Solution

Let y = -1/2"x"+5/2 be the regression line of y on x

and x =-3/2"y" +8/2 be the regression line of x on y

Now, byx=-1/2   "b"_("yx")  = -3/2

sqrt("b"_("yx")."b"_("xy")) = sqrt(( -1)/2.(-3)/2)

=sqrt(3/4)      =(-sqrt3)/2 <1

r =(-sqrt3)/2

Now,   sigma_"x"=sqrt12=2sqrt3

We have:    "b"_("yx") = "r"  sigma_"y"/sigma_"x"

-1/2=-sqrt3/2.sigma_"y"/(2sqrt3)

⇒  sigma_"y"=2

∴ Variance of y =4

coefficient of correlation = (-sqrt3)/2    ...(same sign as "b"_("yx") and "b"_("yx")

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression
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