Recall from the previous quiz the random variables X and Y with joint probability distributionY=0 Y = 1X = 0 0.4 0.1X = 1 0.3 0.2The covariance between X and Y is
Question
Recall from the previous quiz the random variables X and Y with joint probability distributionY=0 Y = 1X = 0 0.4 0.1X = 1 0.3 0.2The covariance between X and Y is
Solution
To calculate the covariance between X and Y, we first need to calculate the expected values of X, Y, XY, and then use the formula for covariance:
Cov(X,Y) = E(XY) - E(X)E(Y)
Step 1: Calculate E(X) and E(Y)
E(X) = ∑ [x * P(X = x)] for all x = (0 * 0.5) + (1 * 0.5) = 0 + 0.5 = 0.5
E(Y) = ∑ [y * P(Y = y)] for all y = (0 * 0.7) + (1 * 0.3) = 0 + 0.3 = 0.3
Step 2: Calculate E(XY)
E(XY) = ∑ ∑ [x * y * P(X = x, Y = y)] for all x and y = (0 * 0 * 0.4) + (0 * 1 * 0.1) + (1 * 0 * 0.3) + (1 * 1 * 0.2) = 0 + 0 + 0 + 0.2 = 0.2
Step 3: Substitute E(X), E(Y), and E(XY) into the formula for Cov(X,Y)
Cov(X,Y) = E(XY) - E(X)E(Y) = 0.2 - (0.5 * 0.3) = 0.2 - 0.15 = 0.05
So, the covariance between X and Y is 0.05.
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