Solution :

Given:

Relation between X and u: 3X + 4u + 7 = 0

Correlation coefficient between X and Y: -0.6


To Find:

Correlation coefficient between u and Y


Steps to Solve:


1. Solve the equation 3X + 4u + 7 = 0 for u

u = (-3/4)X - (7/4)


2. Use the formula for the correlation coefficient between two variables:

r(X,Y) = Cov(X,Y) / (SD(X) * SD(Y))

where Cov(X,Y) is the covariance between X and Y, and SD(X) and SD(Y) are the standard deviations of X and Y, respectively.


3. Substitute u for X in the formula:

r(u,Y) = Cov(u,Y) / (SD(u) * SD(Y))


4. Find the covariance between u and Y:

Cov(u,Y) = Cov((-3/4)X - (7/4), Y)

Cov(u,Y) = (-3/4)Cov(X,Y)

since Cov(aX+b,cY+d) = a*c*Cov(X,Y)


5. Substitute the given correlation coefficient between X and Y:

-0.6 = Cov(X,Y) / (SD(X) * SD(Y))

Cov(X,Y) = -0.6 * SD(X) * SD(Y)


6. Substitute the value of Cov(X,Y) in the expression for Cov(u,Y):

Cov(u,Y) = (-3/4)*(-0.6)*SD(X)*SD(Y)

Cov(u,Y) = 0.45*SD(X)*SD(Y)


7. Substitute the formulas for SD(X) and SD(Y):

SD(X) = sqrt(Var(X))

Var(X) = E[(X-E(X))^2]

SD(Y) = sqrt(Var(Y))

Var(Y) = E[(Y-E(Y))^2]


8. Since E(X) = E(Y) = 0 (given),

Var(X) = E[X^2] and Var(Y) = E[Y^2]


9. Use the relation between X and u to find the variance of u:

Var(u) = Var((-3/4)X - (7/4))

Var(u) = (9/16)Var(X)


10. Substitute the formulas for Var(X) and Var(Y):

Var(X) = E[X^2] = E[(u+(7/4))/(-3/4)]^2 = (16/9)E[u^2]

Var(Y) = E[Y^2]


11. Substitute the formulas for Cov(u,Y), SD(u), and SD(Y) in the expression for r(u,Y):

r(u,Y) = [0.45*SD(X)*SD(Y)] / [SD(u)*SD(Y)]

r(u,Y) = [0.45*sqrt(Var(X))*sqrt(Var