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	0.07
	2	0.06	0.14	0.32

(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

a) By the definition of Conditional Probability we have

So, we have

y	0	1	2

b) By the definition of Conditional Probability we have

So, we have

y	0	1	2

c) Required Probability is P(Y ≤ 1 | X = 2) = 0.1154+0.2692=0.3846

d) By the definition of Conditional Probability we have

So, we have

x	0	1	2