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If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x?
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If the regression coefficient of y on x, the coefficient of correlatio...
Calculating Variance of x when Regression Coefficient, Correlation Coefficient and Variance of y are given
Given:
Formula:
Solution:
Answer: The variance of x is 27/64.
Given:
- Regression coefficient of y on x = -3/4
- Coefficient of correlation between x and y = √3/2
- Variance of y = 4
Formula:
- Regression coefficient of y on x = Covariance(x,y) / Variance(x)
- Coefficient of correlation between x and y = Covariance(x,y) / (Standard deviation of x * Standard deviation of y)
- Variance of y = E[(y - E[y])^2]
Solution:
- Let's first find the covariance between x and y using the regression coefficient formula:
- -3/4 = Covariance(x,y) / Variance(x)
- Covariance(x,y) = (-3/4) * Variance(x)
- Now, let's use the correlation coefficient formula to find the standard deviation of x:
- √3/2 = Covariance(x,y) / (Standard deviation of x * Standard deviation of y)
- Standard deviation of x = Covariance(x,y) / (√3/2 * Standard deviation of y)
- Substituting the value of covariance from the previous equation:
- Standard deviation of x = ((-3/4) * Variance(x)) / (√3/2 * 2)
- Standard deviation of x = (-3/4) * (Variance(x) / √3)
- Now, let's use the variance formula for y:
- 4 = E[(y - E[y])^2]
- Since E[y] = 0 (assuming the mean of y is 0),
- 4 = E[y^2]
- Therefore, Variance(y) = E[y^2] - (E[y])^2 = E[y^2] = 4
- Finally, let's use the variance formula for x:
- Variance(x) = (Standard deviation of x)^2 * 3/4
- Substituting the value of standard deviation of x from the previous equation:
- Variance(x) = ((-3/4) * (Variance(x) / √3))^2 * 3/4
- Simplifying:
- Variance(x) = 27/64
Answer: The variance of x is 27/64.
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If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x?
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If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x?.
If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the regression coefficient of y on x, the coefficient of correlation between x and y and variance of y are -3/4,√3/2 and 4 respectively what is variance of x?.
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