## A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Le

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.
p(x, y) y
0 1 2
x 0 0.10 0.03 0.01
1 0 08 0.20 0.06
2 0.05 0.14 0.33
(a) Given that X = 1, determine the conditional pmf of Y�i.e., pY|X(0|1), pY|X(1|1), pY|X(2|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?
(c) Use the result of part (b) to calculate the conditional probability 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?

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3 years 2021-07-25T12:33:34+00:00 1 Answers 103 views 0

(a): The conditional pmf of Y when X = 1

(b): The conditional pmf of Y when X = 2

(c): From (b) calculate P(Y<=1 | X =2)

(d): The conditional pmf of X when Y = 2

Step-by-step explanation:

Given

The above table

Solving (a): The conditional pmf of Y when X = 1

This implies that we calculate

So, we have:

Reading the data from the given table, the equation becomes

Using the format of the above formula for the rest, we have:

Solving (b): The conditional pmf of Y when X = 2

This implies that we calculate

So, we have:

Reading the data from the given table, the equation becomes

Using the format of the above formula for the rest, we have:

Solving (c): From (b) calculate P(Y<=1 | X =2)

To do this, where Y = 0 or 1

So, we have:

Solving (d): The conditional pmf of X when Y = 2

This implies that we calculate

So, we have:

Reading the data from the given table, the equation becomes

Using the format of the above formula for the rest, we have: