Problem Set 2
Due: Oct. 5 (M), 12:30pm EDT
The exercise question numbers are from Keller, G. (2015). Statistics for Management and
Economics, Abbreviated, 10th ed., Cengage Learning. Review the following information
and learn how to use both the binomial and the (any types of) standard normal tables.
•  ∼ (): () = (1 − )1− for  = 0 1. () =  and () = (1 − ).
•  ∼ B( ): () = ¡

¢
(1 − )− for  = 0 1 ···  , where ¡

¢
= !
!(−)!. () =  and
() = (1 − ).
•  ∼ U[ ]: () = 1
− for  ≤  ≤ . () = +
2 and () = (−)2
12 .
•  ∼ N ¡
 2
¢
: () = √
1
22 −(−)
222
for −∞  ∞. () =  and () = 2.
Part I: Random Variables and Distributions

1. Keller Exercise 7.5.
2. Keller Exercise 7.7. For part (a), draw both the p.m.f. table and the p.m.f. graph.
3. Keller Exercise 7.9.
4. Keller Exercise 7.11.
5. Keller Exercise 7.13. Also draw both the p.m.f. table and the p.m.f. graph.
6. Keller Exercise 7.19. Redo part (d) for another random variable  = −
 , where  is the mean
(i.e., expected value) and 2 is the variance of .
7. Keller Exercise 7.47, 7.48, 7.49, and 7.50.
8. Keller Exercise 7.54 and 7.56.
9. Below is the joint probability table for the discrete random variables  and  . Suppose that
the conditional probability  ( = 1| = 10) is 05.
 = 6  = 8  = 10
 = 1 ?? ? ?
 = 2 0? ? 02
 = 3 02? 02 04
04? 04 1
(a) Fill in the eight “?”s. Recall  ( = | = ) =  ( =   = )  ( = ).
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(b) What is (1 ≤   3)?
(c) What is ( = 8| = 2)?
(d) What is the mean and variance of ?
(e) What is the mean and variance of  ?
(f) What is the correlation coefficient between  and  ?
(g) What is ( +  ) and ( +  )?
(h) Are random variables  and  independent? Why or why not?
Part II: Important Distributions
10. Keller Exercise 7.87 and 7.88.
11. Keller Exercise 7.94.
12. Keller Exercise 7.96.
13. Keller Exercise 7.97.
(a) Let  be the number of people who request a smoking table. What is the distribution of
? You need to specify all parameter values of the distribution.
(b) Answer to (a), (b) and (c).
15. Keller Exercise 8.4.
16. Keller Exercise 8.6.
17. Keller Exercise 8.34, 8.35, and 8.36.
18. Keller Exercise 8.46 and 8.47.
19. Keller Exercise 8.50.
20. Keller Exercise 8.51 and 8.52.
21. Keller Exercise 8.54.
Bonus: Practice with the Standard Normal Table
22. Let  ∼ N (0 1). Using a standard normal table, find the following probabilities. You do not
need to provide any equation. Instead, draw pictures as we did in the lecture and find the
numbers from the table. Make yourself be familiar with using different kinds of tables. [Hint:
The standard normal density is symmetric around zero.]
(a)  ( ≤ 0) (b)  (  196) (c)  ( ≤ 196)
(d)  ( = 196) (e)  (−165  0) (f)  ( ≥ −165)
(g)  ( ≤ −14) (h)  (07 ≤   17) (i)  (−05   ≤ 07)
23. Let  ∼ N (0 1). Using a similar method as Question 15, find the  value in each case.
(a)  ()=005 (b)  ()=0025 (c)  ()=001
(d)  ()=0005 (e)  (0 ≤ )=0475 (f)  ()=0975
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