M362K, Smith, Fall 03

Information for Third Exam

Date: Friday, November 21

Place: ECJ 1.204 ( Not the usual classroom, but the same room as the previous exams.)

Exam will cover: Everything covered in class and homework (reading, practice problems and written problems) through (and including) Friday, November 14. This includes: Bernoulli, binomial, Poisson, geometric, normal, and exponential random variables (what they are; when they are used; their pdf/pmf's, expectations, and variances; which ones are related and how; using them); finding the cmf and pmf of a function of a random variable; joint distributions (joint pmf's, pdf's, cmf's; marginal distributions; using joint pmf's and pdf's to find marginal distributions; using pdf's and pmf's to find probabilities), independent random variables.

Normal tables: You will be provided with a copy of the table on p. 203 giving probabilities for the normal distribution.

Suggestions for studying:

1. Carefully study the first two exams and returned homework to understand things you may have done wrong, including inadequate explanation, improper use of notation, etc. Consult the solutions on reserve in the PMA library as needed.

2. Here are some suggested practice questions and problems.

Please Note:

I. Review Questions

1. Complete the following chart:

Random Variable
 Definition and/or When  It Is Used
 Parameters
 pdf/pmf (formula and graph)
 E(X)
 VAR(X)
 Connections with other random variables
Bernoulli


                




Binomial






Geometric






Poisson






Exponential






Normal







2.If X and Y are discrete random variables, then their joint pmf is defined by:


3.If X and Y are random variables, then the joint cdf of X and Y is defined by:


4. If X and Y are continuous random variables, then their joint cdf is:


5. If X and Y are continuous random variables with joint pdf f, then their joint cdf F can be calculated by:

6. If X and Y are discrete random variable with pmf p, then their joint cdf can be calculated by: 

 
7. If X and Y are jointly distributed, then the distributions of X and Y alone are called _________________ distributions.

8. If X and Y are discrete random variables with joint pmf f(x,y), then the maraginal pmf's of X and Y can be calculated by:
 

9. If X and Y are continuous random variables with joint pdf f(x,y), then the marginal pdf’s of X and Y  can be found by: 

 

10. Random variables X and Y are said to be independent if:
 

II. Review Problems from Textbook


III. Additional Review Problems

1. In each of the following situations, first decide what type of random variable(s) is (are) involved, and explain why. (In particular, state any assumptions you are making.) Then use what you know about the random variable(s) involved to solve the problem.

A. From an ordinary deck of 52 cards, we draw cards at random, with replacement, until a seven is drawn.
 a. What is the probability that exactly  5 draws are needed?
 b. What  is the probability that at least 5 draws are needed?

B. From an ordinary deck of 52 cards, we draw 5 cards at random, with replacement. What is the probability that we get two sevens?
   
C. From an ordinaty deck of 52 cards, we draw a card. You get a dollar if it is a seven, nothing otherwise. What is the probability that you get a dollar?

D. On a certain summer evening, shooting stars are observed  at a rate of one every 12 minutes. What is the probability that three shooting stars are observed in 30 minutes?

E. A restaurant serves  three seafood entrees, four red meat entrees, two poultry entrees, and one vegetarian entree. Suppose customers select from these randomly.
 a. What  is the probability that exactly  two of the next five customers order seafood entrees?
 b. What  is the probability that  none of the next four customers orders a seafood entree, but the fifth does?

F. Suppose that 90% of the patients with a certain disease can be cured with a certain drug. What is the approximate probability that of 50 such patients, at least 45 can be cured with the drug?


2. The joint pmf of the random variables X and Y is given in the following table.


                                                                                             Y


X
1
 2
 3
 4
0
 0
 1/8
 1/8
 1/8
1
 0
 0
 1/8
 1/8
2
 1/8
 1/8
 0
 1/8


    a. Find the marginal pmf of Y.
    b. Find the value of the joint cdf of X and Y at (2,2).

3. X is an exponential random variable with parameter 3. Find the pdf of each of the following random variables:

    a. U = 2X. (What kind of random variable is U? You should be able to tell when you get the pdf.)

    b. Y = X2. (Yes, your answer will be messier than your answer to part (a).)

4. Explain (using pictures, equations, and words as needed to explain your reasoning clearly) how you can find each of the following quantities from the table on p. 203. (Assume Z is a standard normal random variable.)

    a. P{Z ≤ .81}        b. P{Z ≥ .81}    c. P{Z ≤ -.81}    d. P{Z ≥ -.81}    e. P{-.81 ≤ Z ≤ .81}   

    f. C with P{Z ≤ C} = .81    g. C with P{Z ≥ C} = .81    h. C with P{-C ≤ Z ≤ C} = .81

5. You have heard that plain m&m’s contain 30% brown candies. However, you are a peanut m&m’s aficionado, and your extensive experience with peanut m&m’s suggests that they contain fewer than 30% brown, an unfortunate situation, since your prefer brown -- they seem more chocolatey.  Deciding to gather some hard evidence and apply your knowledge of probability to the matter, you carefully keep track of the next 300 peanut m&m’s you consume. You discover that 78 of them are brown. Approximately what is the probability of this happening if the percent of peanut m&m’s which are brown is indeed 30%? Approximately what is the probability of obtaining less than 78 brown m&m's (still assuming the percent of brown is indeed 30%)? Does your answer provide support for your suspicion or not?