Correlation coefficient and regression lines
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A number X is chosen uniformly at random from the numbers 1,2,3,4. After that another number Y is chosen uniformly at random among those at least as large as X. Compute E[X], E[Y], Var(X), Var(Y), Cov(X,Y), the correlation coefficient of X,Y and the regression lines.
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Solution Summary
Computing the correlation coefficient and regression lines from a probability problem. Solution contains a step-by-step information.
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From the condition, we have the following results:
P(X=1) = P(X=2) = P(X=3) = P(X=4) = 1/4
P(Y=1|X=1) = P(Y=2|X=1) = P(Y=3|X=1) = P(Y=4|X=1) = 1/4
P(Y=2|X=2) = P(Y=3|X=2) = P(Y=4|X=2) = 1/3
P(Y=3|X=3) = P(Y=4|X=3) = 1/2
P(Y=4|X=4) = 1
P(Y=1) = P(Y=1|X=1)*P(X=1) = 1/16
P(Y=2) = P(Y=2|X=1)*P(X=1) + P(Y=2|X=2)*P(X=2) = (1/4)(1/4 + 1/3) = 7/48
P(Y=3) = P(Y=3|X=1)*P(X=1) + P(Y=3|X=2)*P(X=2) + P(Y=3|X=3)*P(X=3) = (1/4)(1/4 + 1/3 + 1/2) = 13/48
P(Y=4) = ...
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