Question 1) Let X denote the number of times a certain numerical control machine will malfunction: 0, 1, 2, or 3 times on a given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability mass distribution of X and Y is given by
P(x) = (x + y)/30
for x = 0, 1, 2, 3; and y = 0, 1, 2, find the following:
(a) P( X > 2, Y < 1)
(b) P( X > Y)
(c) P( X + Y = 3)
(d) Marginal of X.
(e) P( Y = 3 I X = 2)
Question 2) The joint density of X and Y is given by
f (x, y) = {(6-x-y)/8, 0 < x < 2, 2< y< 4
0, Otherwise
(a) Find P( X > 1)
(b) Find P( X > 1, Y < 3)
(c) Find P( X + Y < 3 )
Question 3) Two tire-quality experts examine stacks of tires and assign quality ratings to each tire on a 3-point scale. Let X denote the grade given by expert A and Y denote the grade given by B.
The following table gives the joint distribution for X and Y.
f(x,y) y
1 2 3
x 1 0.14 0.05 0.02
2 0.10 0.35 0.05
3 0.03 0.10 0.20
(a) Compute E(X)
(b) Compute E(Y)
(c) Compute E[h(X, Y)] where h[X, Y] = X + Y
(d) Compute Cov(X, Y)
(e) Compute Corr(X, Y)