Question

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation.

		y
p(x, y)		0	1	2
x	0	0.10	0.03	0.02
	1	0.06	0.20	0.08
	2	0.06	0.14	0.31

(a) What is P(X = 1 and Y = 1)?
P(X = 1 and Y = 1) = _____

(b) Compute P(X ≤ 1 and Y ≤ 1).
P(X ≤ 1 and Y ≤ 1) = ______

Compute the probability of this event.
P(X ≠ 0 and Y ≠ 0) =  

(d) Compute the marginal pmf of X.

x	0	1	2
pX(x)

Compute the marginal pmf of Y.

y	0	1	2
pY(y)

Using p_X(x), what is P(X ≤ 1)?
P(X ≤ 1) = ________

Answer 1

a) P(X = 1 and Y = 1) = 0.2

b) P(X < 1 and Y < 1) = P(X = 0 and Y = 0) + P(X = 0 and Y = 1) + P(X = 1 and Y = 0) + P(X = 1 and Y = 1)

                                 = 0.1 + 0.03 + 0.06 + 0.2

                                 = 0.39

P(X ≠ 0 and Y ≠ 0) = P(X = 1 and Y = 1) + P(X = 1 and Y = 2) + P(X = 2 and Y = 1) + P(X = 2 and Y = 2)

                              = 0.2 + 0.08 + 0.14 + 0.31

                              = 0.73

d)

x	P(x)
0	0.1+0.03+0.02 = 0.15
1	0.06+0.2+0.08 = 0.34
2	0.06+0.14+0.31 = 0.51

y	P(y)
0	0.1+0.06+0.06 = 0.22
1	0.03+0.2+0.14 = 0.37
2	0.02+0.08+0.31 = 0.41

P(X < 1)       = P(X = 0) + P(X = 1) = 0.15 + 0.34 = 0.49