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For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the variance of Y if the variance of X is 9. - Mathematics and Statistics

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Question

For bivariate data, the regression coefficient of Y on X is 0.4 and the regression coefficient of X on Y is 0.9. Find the value of the variance of Y if the variance of X is 9.

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Solution

Given: bYX = 0.4, bXY = 0.9,

var(x) = 9; var(y) =?

r = `+-sqrt("b"_"YX"."b"_"XY")`

= `+-sqrt(0.4 xx 0.9)`

= `+-sqrt0.36`

r = 0.6

∵ `"b"_"YX" - "b"_"XY" > 0`

var(x) = 9

`sigma_"X" = sqrt("var(x)")`

= `sqrt9 = 3`

Now, `"b"_"YX" = "r" xx sigma_"Y"/sigma_"X"`

∴ `0.4 = 0.6 xx sigma_"Y"/3`

∴ `0.4 = 0.2 xx sigma_"Y"`

∴ `sigma_"Y" = 0.4/0.2 = 2`

var(y) = `sigma_"y"^2`

= 22 = 4

∴ `sigma^2` = 4

∴  The value of variance of Y is 4.

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Properties of Regression Coefficients
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Chapter 3: Linear Regression - Exercise 3.3 [Page 50]

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