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	0.01
	1	0.07	0.20	0.08
	2	0.06	0.14	0.31

(a) Given that X = 1, determine the conditional pmf of Y—i.e., p_Y|X(0|1), p_Y|X(1|1), p_Y|X(2|1). (Round your answers to four decimal places.)

y	0	1	2
pY|X(y|1)

(b) Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island? (Round your answers to four decimal places.)

y	0	1	2
pY|X(y|2)

(c) Use the result of part (b) to calculate the conditional probability P(Y ≤ 1 | X = 2). (Round your answer to four decimal places.)
P(Y ≤ 1 | X = 2) =  

(d) Given that two hoses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island? (Round your answers to four decimal places.)

x	0	1	2
pX|Y(x|2)

Answer 1

Following table shows the joint and marginal pdf:

			Y
	p(x,y)	0	1	2	P(X=x)
	0	0.1	0.03	0.01	0.14
X	1	0.07	0.2	0.08	0.35
	2	0.06	0.14	0.31	0.51
	P(Y=y)	0.23	0.37	0.40	1

(a)

The required conditional pdf is:

Y	0	1	2
P(Y|X=1)	0.07/0.35=0.20	0.2/0.35=0.5714	0.08/0.35=0.2286

(b)

The required conditional pdf is:

Y	0	1	2
P(Y|X=2)	0.06/0.51=0.1176	0.14/0.51=0.2745	0.31/0.51=0.6078

(c)

(D)

X	0	1	2
P(X|Y=2)	0.01/0.40=0.025	0.08/0.40=0.20	0.31/0.40=0.775