Question

if the regression coefficient of Y on X ,the coefficient of correlation between x and y and variance of y are 3 /4 , root 3 by 2, and 4 respectively what is the variance of x?​

Answer 1

Given:

The regression coefficient of Y on X, the coefficient of correlation between x and y and variance of y are 3 /4 , root 3 by 2, and 4 respectively

To find:

What is the variance of x?​

Solution:

From given, we have,

The regression coefficient of Y on X = b_{yx} = 3/4

The coefficient of correlation between x and y = r = √(3/2 )

The variance of y = V(y) = 4

⇒ The standard deviation of y = S(y) = 2

we use the formula,

r² = b_{yx} × b_{xy}

√(3/2)² = 3/4 × b_{xy}

3/2 = 3/4 × b_{xy}

∴ b_{xy} = 2

where, b_{xy} is regression coefficient of X on Y

Now we use the formula,

b_{xy} = r × S(x)/S(y)

2 = √(3/2 ) × S(x)/2

4 = √(3/2 ) × S(x)

S(x) = 4√(2/3 )

∴ V(x) = [4√(2/3 )]² = 32/3

Therefore, the variance of x is 32/3