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

22 −(−)

222

for −∞ ∞. () = and () = 2.

Part I: Random Variables and Distributions

- Keller Exercise 7.5.
- Keller Exercise 7.7. For part (a), draw both the p.m.f. table and the p.m.f. graph.
- Keller Exercise 7.9.
- Keller Exercise 7.11.
- Keller Exercise 7.13. Also draw both the p.m.f. table and the p.m.f. graph.
- 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 . - Keller Exercise 7.47, 7.48, 7.49, and 7.50.
- Keller Exercise 7.54 and 7.56.
- Below is the joint probability table for the discrete random variables and . Suppose that

the conditional probability ( = 1| = 10) is 05.

= 6 = 8 = 10

= 1 ?? ? ?

= 2 0? ? 02

= 3 02? 02 04

04? 04 1

(a) Fill in the eight “?”s. Recall ( = | = ) = ( = = ) ( = ).

1

(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 - Keller Exercise 7.87 and 7.88.
- Keller Exercise 7.94.
- Keller Exercise 7.96.
- Keller Exercise 7.97.
- Read Keller Exercise 7.127.

(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). - Keller Exercise 8.4.
- Keller Exercise 8.6.
- Keller Exercise 8.34, 8.35, and 8.36.
- Keller Exercise 8.46 and 8.47.
- Keller Exercise 8.50.
- Keller Exercise 8.51 and 8.52.
- Keller Exercise 8.54.

Bonus: Practice with the Standard Normal Table - 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) ( 196) (c) ( ≤ 196)

(d) ( = 196) (e) (−165 0) (f) ( ≥ −165)

(g) ( ≤ −14) (h) (07 ≤ 17) (i) (−05 ≤ 07) - Let ∼ N (0 1). Using a similar method as Question 15, find the value in each case.

(a) ()=005 (b) ()=0025 (c) ()=001

(d) ()=0005 (e) (0 ≤ )=0475 (f) ()=0975

2