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In a certain community, levels of air pollution may exceed

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 1E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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1
Problem 1E

In a certain community, levels of air pollution may exceed federal standards for ozone or for particulate matter on some days. For a particular summer season, let X be the number of days on which the ozone standard is exceeded and let Y be the number of days on which the particulate matter standard is exceeded. Assume that the joint probability mass function of X and Y is given in the following table:

 

Y

X

0

1

2

0

0.10

0.11

0.05

1

0.17

0.23

0.08

2

0.6

014

0.06

a. Find P(X = 1 and Y = 0)

b. Find P(X  > 1 and Y < 2).

c. Find P(X < 1).

d. Find P(Y≥ 1).

e. Find the probability that the standard for ozone is exceeded at least once.

f. Find the probability that the standard for particulate matter is never exceeded.

g. Find the probability that neither standard is ever exceeded.

Step-by-Step Solution:

Answer :

Step 1 of 8: 

            Given, in a certain community, levels of air pollution may exceed federal standards for ozone or for particulate matter on some days. The joint probability mass function of X and Y is in the below table.

                

             

                           Y

     X

           0

           1

        2

     0

          0.10

          0.11

        0.05

     1

          0.17

          0.23

        0.08

     2

          0.06

          0.14

         0.06

  Here, X be the number of days on which the ozone standard is exceeded.

            Y be the number of days on which the particulate matter standard exceeded.

       The joint probability mass function of two discrete random variables X and Y is defined as

                                           

Step 2 of 8:

 a). To find P(X = 1 and Y = 0).

                                              From the above table,

          P(X = 1 and Y = 0) = 0.17.

                                             Therefore,

                   P(X = 1 and Y = 0) = 0.17.

 Step 3 of 8:

  b). To find

       Then,

                =

   

     = 0.17+0.23+0.06+0.14.

                                                             = 0.6                                  

                Therefore, = 0.6.

 Step 4 of 8:

c). To find

       Then,

                 

                       = 0.10+0.11+0.05

    = 0.26

        Therefore, = 0.26.

Step 5 of 8

Chapter 2.6, Problem 1E is Solved
Step 6 of 8

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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