A service station has both self-service and full-service islands. On each island, there is a single regular...

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A service station has both self-service and full-service islands. On each island, there is a single regular...

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	0.08	0.20	0.06
	2	0.05	0.14	0.33

(a) Given that X = 1, determine the conditional pmf of Y�i.e., 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
p_Y\|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
p_Y\|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
p_X\|Y(x\|2)

math Statistics-And-Probability

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Answer 1

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Answer #1

Concepts and reason

Joint probability mass function: If two or more events occur together or at same point of time, then the probability of the two events is called as joint probability. In other words, the probability of the intersection of two events is defined as joint probability. The sum of the probability of different outcomes in the probability distribution must add up to one.

Probability: The ratio of the number of favorable outcomes to certain event and total number of possible outcomes is called as the probability of an event.

Conditional probability: The probability of happening of an event given that another event has already happened is called as the conditional probability.

Fundamentals

The conditional probability mass function of the random variables X given Y is,

Pax (y|x) = P(x,y)
Px(x)

The conditional probability mass function of the random variables Y given X is,

$Prix (X\Y) = P(x,y)$

(a)

p(x, y)
0
1
1
0
2
Total
0.10
.08
0.05
0.23
0.03
0.20
0.14
0.37
2 Total
0.01 10.14
0.06 0.34
0.33 0.52
0.40 1

The conditional probability mass function of given is,

Prix (011) - P(1,0)
P: (1)
0.08
0.34
= 0.2353

The conditional probability mass function of given is,

Prix (1/1) - P(1,1)
Px (1)
0.20
0.34
= 0.5882

The conditional probability mass function of given is,

Prix (211) =
P(1,2)
Px(1)
0.06
0.34
= 0.1765

(b)

The conditional probability mass function of given X = 2 is,

_P(2,0)
Prix (012) =
Px (2)
0.05
0.52
= 0.0962

The conditional probability mass function of given X = 2 is,

Prix (1/2) = P(2,1)
Px (2)
0.14
0.52
= 0.2692

The conditional probability mass function of given X = 2 is,

Prix (2/2) - P(2,2)
Px (2)
0.33
0.52
= 0.6346

(c)

The conditional probability of given X = 2 is,

P(Y 31|X = 2) = Prix (012) + Prix (112)
= 0.0962 +0.2692
= 0.3654

(d)

The conditional probability mass function of X = 0 given is,

The conditional probability mass function of given is,

Prir (1/2) - P(1,2)
P,(2)
0.06
0.40
= 0.1500

The conditional probability mass function of given is,

P012) P(2,2)
Pxly (212) =
P, (2)
0.33
0.40
= 0.8250

Ans: Part a

The conditional probability mass function of given is as follows:

Part b

The conditional probability mass function of, and given is as follows:

Part c

The conditional probability of given is 0.3654.

Part d

The conditional probability mass function of, and given is as follows:

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Answer 2

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