Suppose the joint probability distribution of X and Y is givenby the following table.
Y=>3 6 9 X
1 0.2 0.2 0
2 0.2 0 0.2
3 0 0.1 0.1
The table entries represent the probabilities. Hence theoutcome [X=1,Y=6] has probability
0.2.
a) Compute E(X), E(X2), E(Y), and E(XY). (For all answers showyour work.) b) Compute E[Y | X = 1], E[Y | X = 2], and E[Y | X =3].
c) In this case, E[Y | X] is linear, given by E[Y | X] = ?0 +?1X where ?0 and ?1 are constants. Make a plot with E[Y | X] on thevertical axis and X on the horizontal. Can you use your plot todeduce the values of ?0 and ?1?
d) When E[Y | X] is linear, a formula for ?1 is ?1 =Cov(X,Y)/Var(X).
And given ?1, a formula for ?0 is ?0 = E(Y) – ?1E(X).
Does applying these formulas yield the same answers that youdeduced in part c?
e) Let u = Y – (?0 + ?1X). It so happens that u can take fourpossible values: 1.5, -1.5, 3, and -3. Find the joint distributionof u and X. The first row is done for you.
u=>-3 -1.5 1.5 3 X
1 0 0.2 0.2 0
2 _____ _____ _____ _____
3 _____ _____ _____ _____
Does E(u) = 0? Does Cov(X,u) = 0?
