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	0.05	0.02
	1	0.07	0.20	0.08
	2	0.06	0.14	0.28

(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

From the table we know the following joint probabilities

We also find the following marginal probabilities by getting the row and column sums

p(x, y)		0	1	2	Marginal probabilities
x	0	0.10	0.05	0.02	0.17
	1	0.07	0.2	0.08	0.35
	2	0.06	0.14	0.28	0.48
Marginal probabilities		0.23	0.39	0.38	1.00

The marginal probabilities are

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).

Using the formula for conditional probabilities we get

ans:

y	0	1	2
pY|X(y|1)	0.2000	0.5714	0.2286

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?

ans:

y	0	1	2
pY|X(y|2)	0.1250	0.2917	0.5833

c)  the conditional probability P(Y ≤ 1 | X = 2) is

ans: the conditional probability P(Y ≤ 1 | X = 2) is 0.4167

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?

ans:

x	0	1	2
pX|Y(x|2)	0.0526	0.2105	0.7368