Suppose X and Y are random variables such that X follows a discrete uniform distribution on the set {0, 1} and Y = X2. Determine Cov (X, Y). Enter the answer correct to two decimal places.
Question
Suppose X and Y are random variables such that X follows a discrete uniform distribution on the set {0, 1} and Y = X2. Determine Cov (X, Y). Enter the answer correct to two decimal places.
Solution
To find the covariance between X and Y, we use the formula:
Cov(X,Y) = E[XY] - E[X]E[Y]
Given that X follows a discrete uniform distribution on the set {0, 1}, the expected value E[X] is (0+1)/2 = 0.5.
Since Y = X^2, for X = 0 and 1, Y also takes the values 0 and 1. So, E[Y] is also (0+1)/2 = 0.5.
Now, we need to find E[XY]. But since Y = X^2, XY = X*X^2 = X^3. The possible values of X are 0 and 1, so the possible values of X^3 are also 0 and 1. Therefore, E[XY] = E[X^3] = (0+1)/2 = 0.5.
Now we can substitute these values into the covariance formula:
Cov(X,Y) = E[XY] - E[X]E[Y] = 0.5 - 0.5*0.5 = 0.5 - 0.25 = 0.25.
So, the covariance between X and Y is 0.25.
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