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If the covariance between two variables is 20 and the variance of one of the variables is 16, what would be the variance of the variables is 16, what would be the variance of the other variable?
- a)More than 10
- b)More than 100
- c)More than 1.25
- d)Lss than 10
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If the covariance between two variables is 20 and the variance of one ...
Explanation:
Let X and Y be two variables.
Covariance between two variables is denoted by Cov(X, Y) = 20.
Variance of one of the variables is denoted by Var(X) = 16.
Variance of the other variable is denoted by Var(Y).
To find Var(Y), we need to use the formula of covariance:
Cov(X, Y) = E[(X - μx)(Y - μy)]
where μx and μy are the means of X and Y respectively.
We can write the above formula as:
Cov(X, Y) = E[XY] - μxE[Y] - μyE[X] + μxμy
Since we do not have the means of X and Y, we can use the following formulas:
E[X] = μx and E[Y] = μy
Therefore, the above formula becomes:
Cov(X, Y) = E[XY] - E[X]E[Y]
Substituting the given values in the formula, we get:
20 = E[XY] - 4E[Y]
E[XY] = 20 + 4E[Y]
Now, we can use the formula of variance:
Var(X) = E[X^2] - (E[X])^2
Substituting the given values in the formula, we get:
16 = E[X^2] - 16
E[X^2] = 32
Similarly, we can find:
E[Y^2] = 104
Now, we can use the formula of variance:
Var(Y) = E[Y^2] - (E[Y])^2
Substituting the given values in the formula, we get:
Var(Y) = 104 - (E[Y])^2
Using the formula of covariance, we can write:
20 = E[XY] - 4E[Y]
E[XY] = 20 + 4E[Y]
Substituting the above value in the formula of covariance, we get:
20 = (20 + 4E[Y]) - 4E[Y]
20 = 20
Therefore, the above equation is always true.
Now, we can use the formula of correlation:
r(X, Y) = Cov(X, Y) / (σx * σy)
Since we know Cov(X, Y) and Var(X), we can find σx:
σx = sqrt(Var(X)) = sqrt(16) = 4
Therefore, we can write:
r(X, Y) = 20 / (4 * σy)
Since the correlation coefficient is between -1 and 1, we can say:
-1 ≤ r(X, Y) ≤ 1
Therefore, we can write:
-1 ≤ 20 / (4 * σy) ≤ 1
-4 ≤ σy ≤ 4
Since the variance cannot be negative, we can say:
σy ≥ 0
Therefore, the possible values of σy
Let X and Y be two variables.
Covariance between two variables is denoted by Cov(X, Y) = 20.
Variance of one of the variables is denoted by Var(X) = 16.
Variance of the other variable is denoted by Var(Y).
To find Var(Y), we need to use the formula of covariance:
Cov(X, Y) = E[(X - μx)(Y - μy)]
where μx and μy are the means of X and Y respectively.
We can write the above formula as:
Cov(X, Y) = E[XY] - μxE[Y] - μyE[X] + μxμy
Since we do not have the means of X and Y, we can use the following formulas:
E[X] = μx and E[Y] = μy
Therefore, the above formula becomes:
Cov(X, Y) = E[XY] - E[X]E[Y]
Substituting the given values in the formula, we get:
20 = E[XY] - 4E[Y]
E[XY] = 20 + 4E[Y]
Now, we can use the formula of variance:
Var(X) = E[X^2] - (E[X])^2
Substituting the given values in the formula, we get:
16 = E[X^2] - 16
E[X^2] = 32
Similarly, we can find:
E[Y^2] = 104
Now, we can use the formula of variance:
Var(Y) = E[Y^2] - (E[Y])^2
Substituting the given values in the formula, we get:
Var(Y) = 104 - (E[Y])^2
Using the formula of covariance, we can write:
20 = E[XY] - 4E[Y]
E[XY] = 20 + 4E[Y]
Substituting the above value in the formula of covariance, we get:
20 = (20 + 4E[Y]) - 4E[Y]
20 = 20
Therefore, the above equation is always true.
Now, we can use the formula of correlation:
r(X, Y) = Cov(X, Y) / (σx * σy)
Since we know Cov(X, Y) and Var(X), we can find σx:
σx = sqrt(Var(X)) = sqrt(16) = 4
Therefore, we can write:
r(X, Y) = 20 / (4 * σy)
Since the correlation coefficient is between -1 and 1, we can say:
-1 ≤ r(X, Y) ≤ 1
Therefore, we can write:
-1 ≤ 20 / (4 * σy) ≤ 1
-4 ≤ σy ≤ 4
Since the variance cannot be negative, we can say:
σy ≥ 0
Therefore, the possible values of σy
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If the covariance between two variables is 20 and the variance of one of the variables is 16, what would be the variance of the variables is 16, what would be the variance of the other variable?a)More than 10b)More than 100c)More than 1.25d)Lss than 10Correct answer is option 'B'. Can you explain this answer?
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