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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 other variable?
- a)More than 100
- b)More than 10
- c)Less than 10
- d)More than 1.25
Correct answer is option 'A'. 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:
Covariance:
Covariance is a measure of the joint variability of two random variables. It measures how much two variables change together.
Formula:
Covariance(X,Y) = E[(X - E[X]) * (Y - E[Y])]
Given:
Covariance(X,Y) = 20
Variance(X) = 16
Find:
Variance(Y)
Using Formula:
Covariance(X,Y) = E[(X - E[X]) * (Y - E[Y])]
=> 20 = E[(X - E[X]) * (Y - E[Y])]
Variance(X) = E[(X - E[X])^2]
=> 16 = E[(X - E[X])^2]
Substituting:
20 = E[(X - E[X]) * (Y - E[Y])]
=> 20 = E[XY - XE[Y] - YE[X] + E[X]E[Y]]
=> 20 = E[XY] - E[X]E[Y] - E[Y]E[X] + E[X]E[Y]
=> 20 = E[XY] - E[X]E[Y]
16 = E[(X - E[X])^2]
=> 16 = E[X^2 - 2XE[X] + E[X]^2]
=> 16 = E[X^2] - 2E[X]E[X] + E[X]^2
=> 16 = E[X^2] - E[X]^2
Simplifying:
E[XY] = Covariance(X,Y) + E[X]E[Y] = 20 + E[X]E[Y]
E[X^2] = Variance(X) + E[X]^2 = 16 + E[X]^2
Using Formula:
Variance(Y) = E[(Y - E[Y])^2]
=> Variance(Y) = E[Y^2 - 2YE[Y] + E[Y]^2]
=> Variance(Y) = E[Y^2] - 2E[Y]E[Y] + E[Y]^2
=> Variance(Y) = E[Y^2] - E[Y]^2
Substituting:
Variance(Y) = (E[XY] - E[X]E[Y]) - E[Y]^2
=> Variance(Y) = (20 + E[X]E[Y]) - E[X]^2 - 2E[X]E[Y] - E[Y]^2
=> Variance(Y) = 20 - E[X]^2 - E[Y]^2 + E[X]E[Y]
Substituting:
Variance(Y) = 20 - 16 - E[Y]^2 + E[X]E[Y]
=> Variance(Y) = 4 + E[X]E[Y] - E[Y]^2
As we don't know the value of E[X]E[Y] and E[Y]^2, we can't calculate the exact value of Variance(Y).
However, we can say that Variance(Y) will be more than 100 because E[X]E[Y] will be positive and E[Y]^2 will be less than or equal to 0 (as variance can
Covariance:
Covariance is a measure of the joint variability of two random variables. It measures how much two variables change together.
Formula:
Covariance(X,Y) = E[(X - E[X]) * (Y - E[Y])]
Given:
Covariance(X,Y) = 20
Variance(X) = 16
Find:
Variance(Y)
Using Formula:
Covariance(X,Y) = E[(X - E[X]) * (Y - E[Y])]
=> 20 = E[(X - E[X]) * (Y - E[Y])]
Variance(X) = E[(X - E[X])^2]
=> 16 = E[(X - E[X])^2]
Substituting:
20 = E[(X - E[X]) * (Y - E[Y])]
=> 20 = E[XY - XE[Y] - YE[X] + E[X]E[Y]]
=> 20 = E[XY] - E[X]E[Y] - E[Y]E[X] + E[X]E[Y]
=> 20 = E[XY] - E[X]E[Y]
16 = E[(X - E[X])^2]
=> 16 = E[X^2 - 2XE[X] + E[X]^2]
=> 16 = E[X^2] - 2E[X]E[X] + E[X]^2
=> 16 = E[X^2] - E[X]^2
Simplifying:
E[XY] = Covariance(X,Y) + E[X]E[Y] = 20 + E[X]E[Y]
E[X^2] = Variance(X) + E[X]^2 = 16 + E[X]^2
Using Formula:
Variance(Y) = E[(Y - E[Y])^2]
=> Variance(Y) = E[Y^2 - 2YE[Y] + E[Y]^2]
=> Variance(Y) = E[Y^2] - 2E[Y]E[Y] + E[Y]^2
=> Variance(Y) = E[Y^2] - E[Y]^2
Substituting:
Variance(Y) = (E[XY] - E[X]E[Y]) - E[Y]^2
=> Variance(Y) = (20 + E[X]E[Y]) - E[X]^2 - 2E[X]E[Y] - E[Y]^2
=> Variance(Y) = 20 - E[X]^2 - E[Y]^2 + E[X]E[Y]
Substituting:
Variance(Y) = 20 - 16 - E[Y]^2 + E[X]E[Y]
=> Variance(Y) = 4 + E[X]E[Y] - E[Y]^2
As we don't know the value of E[X]E[Y] and E[Y]^2, we can't calculate the exact value of Variance(Y).
However, we can say that Variance(Y) will be more than 100 because E[X]E[Y] will be positive and E[Y]^2 will be less than or equal to 0 (as variance can
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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 other variable?a)More than 100b)More than 10c)Less than 10d)More than 1.25Correct answer is option 'A'. Can you explain this answer?
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