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प्रश्न
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.
उत्तर
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.
संबंधित प्रश्न
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| x | y | `x - barx` | `y - bary` | `(x - barx)(y - bary)` | `(x - barx)^2` | `(y - bary)^2` |
| 1 | 5 | – 2 | – 4 | 8 | 4 | 16 |
| 2 | 7 | – 1 | – 2 | `square` | 1 | 4 |
| 3 | 9 | 0 | 0 | 0 | 0 | 0 |
| 4 | 11 | 1 | 2 | 2 | 4 | 4 |
| 5 | 13 | 2 | 4 | 8 | 1 | 16 |
| Total = 15 | Total = 45 | Total = 0 | Total = 0 | Total = `square` | Total = 10 | Total = 40 |
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| x | y | xy | x2 | y2 |
| 6 | 9 | 54 | 36 | 81 |
| 2 | 11 | 22 | 4 | 121 |
| 10 | 5 | 50 | 100 | 25 |
| 4 | 8 | 32 | 16 | 64 |
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