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Contracts for two construction jobs are randomly assigned

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer ISBN: 9780495110811 47

Solution for problem 1E Chapter 5

Mathematical Statistics with Applications | 7th Edition

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Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

Mathematical Statistics with Applications | 7th Edition

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Problem 1E

Contracts for two construction jobs are randomly assigned to one or more of three firms, A, B, and C. Let Y1 denote the number of contracts assigned to firm A and Y2 the number of contracts assigned to firm B. Recall that each firm can receive 0, 1, or 2 contracts.

a Find the joint probability function for Y1 and Y2.

b Find F(1, 0).

Step-by-Step Solution:

Answer:

Step 1 of 2:

(a)

The contract for two construction jobs are randomly assigned to one or more of three firms,  Let  denote the number of contracts assigned to firm  and  the number of contracts assigned to firm

Recall that each firm can receive  contracts.

We need to find the joint probability function for  and

Let  and  be discrete random variables. The joint (or bivariate) probability function for  and  is given by,

…………(1)

Let  denote the pair of event that  contracts are assigned to firm  and  contracts are assigned to firm  

Here  can take values  since there are only two construction jobs.

To find the total number of events in the sample space, we need to find out the number of pairs.

So by the product rule, a total number of ways is

Therefore the sample space

 

 

Hence the probability that  contracts is assigned to firm  and 0  contracts assigned to firm

Using equation (1),

              [ ]

Similarly, we can write,

           

           

Similarly, we can write the probability,

Lets update the above values in the joint table.

Hence the joint distribution of

   \  

0

1

2

Total

0

1

0

2

0

0

Total

1

Step 2 of 2

Chapter 5, Problem 1E is Solved
Textbook: Mathematical Statistics with Applications
Edition: 7
Author: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer
ISBN: 9780495110811

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