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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 the variance of x?
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If the regression coefficient of y on X, the coefficient of correlatio...
Given:
- Regression coefficient of y on x (β): 3/4
- Coefficient of correlation between x and y (r): √3/2
- Variance of y (σy^2): 4
Find:
Variance of x (σx^2)
Solution:
1. Formula for Variance of y:
The variance of y can be calculated using the formula:
σy^2 = β^2 * σx^2 + σ^2
2. Substituting Given Values:
Given that σy^2 = 4 and β = 3/4, we can substitute these values into the formula:
4 = (3/4)^2 * σx^2 + σ^2
3. Formula for Correlation Coefficient:
The coefficient of correlation (r) is related to the regression coefficient (β) by the formula:
r = β * (σx / σy)
4. Substituting Given Values:
Given that r = √3/2 and β = 3/4, we can substitute these values into the formula:
√3/2 = 3/4 * (σx / 2)
5. Solving for Variance of x:
From step 4, we can solve for σx:
√3/2 = 3/4 * (σx / 2)
σx = (√3/2) * (4/3)
σx = 2√3 / 3
6. Calculating Variance of x:
To find the variance of x (σx^2), we square the value of σx:
σx^2 = (2√3 / 3)^2
σx^2 = 4/3
7. Conclusion:
Therefore, the variance of x is 4/3.
- Regression coefficient of y on x (β): 3/4
- Coefficient of correlation between x and y (r): √3/2
- Variance of y (σy^2): 4
Find:
Variance of x (σx^2)
Solution:
1. Formula for Variance of y:
The variance of y can be calculated using the formula:
σy^2 = β^2 * σx^2 + σ^2
2. Substituting Given Values:
Given that σy^2 = 4 and β = 3/4, we can substitute these values into the formula:
4 = (3/4)^2 * σx^2 + σ^2
3. Formula for Correlation Coefficient:
The coefficient of correlation (r) is related to the regression coefficient (β) by the formula:
r = β * (σx / σy)
4. Substituting Given Values:
Given that r = √3/2 and β = 3/4, we can substitute these values into the formula:
√3/2 = 3/4 * (σx / 2)
5. Solving for Variance of x:
From step 4, we can solve for σx:
√3/2 = 3/4 * (σx / 2)
σx = (√3/2) * (4/3)
σx = 2√3 / 3
6. Calculating Variance of x:
To find the variance of x (σx^2), we square the value of σx:
σx^2 = (2√3 / 3)^2
σx^2 = 4/3
7. Conclusion:
Therefore, the variance of x is 4/3.
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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 the 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 the variance of x? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC 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 the variance of x? covers all topics & solutions for UPSC 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 the 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 the variance of x? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC 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 the variance of x? covers all topics & solutions for UPSC 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 the variance of x?.
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