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).

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 |
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1 |
0 |
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2 |
0 |
0 |
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Total |
1 |