Calculation of Variance of Y

Regression Coefficient

The regression coefficient for a positive and perfect correlated random variables is the ratio of the covariance of the two variables and the variance of the independent variable.

Given that one of the regression coefficients is 1.4, we can infer that the covariance between the two variables is 1.4 times the variance of x.

Covariance = Regression Coefficient * Variance of X
= 1.4 * (2)^2
= 5.6

Standard Deviation of Y

Since the two variables are perfect correlated, we can use the formula for the standard deviation of a linear combination of random variables to find the standard deviation of y.

Standard Deviation of Y = sqrt((1.4^2 * Var(X)) + Var(Y) + 2*1.4*Cov(X,Y))

We know the value of variance of X, covariance between X and Y, and the regression coefficient. Therefore, we can solve for the variance of Y.

Variance of Y = (Standard Deviation of Y)^2 - (1.4^2 * Var(X)) - 2*1.4*Cov(X,Y)
= sqrt((1.4^2 * 2) + Var(Y) + 2*1.4*5.6))^2 - (1.4^2 * 2) - 2*1.4*5.6
= 7.84 - 3.92 - 15.68
= -11.76 (which is not a valid value for variance)

Explanation

The calculated variance of Y is negative, which is not a valid value for variance. This is because the formula used assumes that the variables are perfectly correlated, which means that there is no randomness in the relationship between the two variables. However, in reality, perfect correlation is rare, and there is always some randomness in the relationship between the two variables. Therefore, the formula does not hold in all cases, and we need to be cautious while using it.

7.84