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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 ...
Solution:
Given, Covariance between two variables = 20
Variance of one of the variables = 16
Let the two variables be X and Y.
Covariance formula is given as:
Cov(X, Y) = E[(X - μX)(Y - μY)]
where E is the expected value, μX and μY are the means of X and Y respectively.
We can rewrite the covariance formula as:
Cov(X, Y) = E[XY] - μXμY
Now, we can use this formula to find the variance of the other variable.
Variance formula is given as:
Var(Y) = E[(Y - μY)²]
We can rewrite the variance formula as:
Var(Y) = E[Y²] - μY²
Now, we can use the covariance formula and the variance formula to find Var(Y).
Var(Y) = Cov(X, Y) + μXμY (from covariance formula)
We know that Cov(X, Y) = 20 and Var(X) = 16.
Let us assume that μX = μY = μ (for simplicity).
Then, we can write:
Var(X) = E[X²] - μ² (from variance formula)
16 = E[X²] - μ²
E[X²] = 16 + μ²
Similarly, we can write:
Cov(X, Y) = E[XY] - μ² (from covariance formula)
20 = E[XY] - μ²
E[XY] = 20 + μ²
Now, we can substitute these values in the expression for Var(Y).
Var(Y) = Cov(X, Y) + μXμY
Var(Y) = 20 + μ² + μ²
Var(Y) = 2μ² + 20
We can see that Var(Y) depends on μ², which can take any positive value.
Therefore, Var(Y) can be more than 100.
Hence, option A is the correct answer.
Given, Covariance between two variables = 20
Variance of one of the variables = 16
Let the two variables be X and Y.
Covariance formula is given as:
Cov(X, Y) = E[(X - μX)(Y - μY)]
where E is the expected value, μX and μY are the means of X and Y respectively.
We can rewrite the covariance formula as:
Cov(X, Y) = E[XY] - μXμY
Now, we can use this formula to find the variance of the other variable.
Variance formula is given as:
Var(Y) = E[(Y - μY)²]
We can rewrite the variance formula as:
Var(Y) = E[Y²] - μY²
Now, we can use the covariance formula and the variance formula to find Var(Y).
Var(Y) = Cov(X, Y) + μXμY (from covariance formula)
We know that Cov(X, Y) = 20 and Var(X) = 16.
Let us assume that μX = μY = μ (for simplicity).
Then, we can write:
Var(X) = E[X²] - μ² (from variance formula)
16 = E[X²] - μ²
E[X²] = 16 + μ²
Similarly, we can write:
Cov(X, Y) = E[XY] - μ² (from covariance formula)
20 = E[XY] - μ²
E[XY] = 20 + μ²
Now, we can substitute these values in the expression for Var(Y).
Var(Y) = Cov(X, Y) + μXμY
Var(Y) = 20 + μ² + μ²
Var(Y) = 2μ² + 20
We can see that Var(Y) depends on μ², which can take any positive value.
Therefore, Var(Y) can be more than 100.
Hence, option A is the correct answer.
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Community Answer
If the covariance between two variables is 20 and the variance of one ...
Given, Cov (x, y) =20
and variance of one of the variables is 16.
so the standard deviation SD is 4.
we know the formula,
r= cov (x, y) /(SD of x * SD of Y)
r= 20 /4 * SD of the other variable
r= 5/ SD of the other variable
we also know that coefficient of correlation, r, lies between -1 and +1 including them.
so, SD of the other variable has to be atleast 5 or more.
so the variance will be 5^2 = 25 atleast or more.
i dont know why the icai book says the answer is more than 100. if anybody can match the answer with book then please tell me.
and variance of one of the variables is 16.
so the standard deviation SD is 4.
we know the formula,
r= cov (x, y) /(SD of x * SD of Y)
r= 20 /4 * SD of the other variable
r= 5/ SD of the other variable
we also know that coefficient of correlation, r, lies between -1 and +1 including them.
so, SD of the other variable has to be atleast 5 or more.
so the variance will be 5^2 = 25 atleast or more.
i dont know why the icai book says the answer is more than 100. if anybody can match the answer with book then please tell me.
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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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