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Homework #3 - Discrete Random Variables and Probability Distributions

The third homework covers section 8 of chapter 3 and all of chapter 4 (i.e., 3.8, 4.1-4.9).  The materials in those sections imply the following skills as being important: 

1. Identifying a random variable and the range of outcomes.
2. Identifying the probability density function (pdf) and cumulative density function (cdf) for a discrete random variable, and verifying that the pdf and/or cdf properties hold.
3. Manipulating the pdf and cdf to answer probability questions (e.g., P(X<5), P(X>=7.5), P(3<X<9)).
4. Identifying and recognizing random variables that conform to the definitions of the six "standard" random variables - uniform, binomial, geometric, negative binomial, hypergeometric, and poisson. 
5. Manipulating the pdf and cdf of the six standard discrete random variable formulations to determine probabilities, sample size, or proportions satisfying certain conditions.  

Assignment

Book Problems: 

1. 4-93
2. 4-94
3. 4-95
4. 4-96
5. 4-104
   Also add the following parts, in which you should calculate the request value:
   (b).  P(X <=  6)
   (c).  P(X > 8)
   (d).  P(3 <= X <= 7)
   (e).  E(X)
   (f).  Var(X)
   
6. 4-113
7. 4-114
8. 4-116
9. 4-117

Additional Problems

10.   Identifying Random Variables in Your Discipline:  The purpose of this problem is to encourage you to think about what constitutes a  random variable in your discipline.
    
    (a). State your disipline (major)
    
    For each of the next parts of the problem, you should (1) identify a discipline-relevant random variable with the properties requested and (2) describe the range of the random variable:
    
    (b). 1 continuous random variable
    (c). 1 discrete random variable
            (d). 1 uniform discrete random variable
            (e). 1 binomial  random variable
            (f).  1 geometric  random variable
            (g). 1 negative binomial random variable
            (h). 1 hypergeometric random variable
            (i).  1 poisson random variable
    
    Hint:  To gain guidance on this problem, you might look to problems 3-83 through 3-91 for ideas about distinguishing between discrete and continuous random variables.  You might look to problems 4-1 through 4-10 for ideas about determining the range of the random variable.  You might look to the examples and problems in sections 4-5 through 4-9 for ideas about identifying random variables for each of the "standard" discrete distributions.  Finally, in determining random variables for the binomial, geometric, negative binomial, and hypergeometric distributions, you might consider the relationships among the definitions of the random variables for these distributions.
    
11.   Lottery Problem:  Lotto is a popular game run by the State of Washington.  Total lottery sales for 1995 were approximately $400 million, whereof 51% are paid out in prizes.  Of the state revenue (which is 35% of the total sales), 60% are used to support the public schools (from "Winning the beat").  In this problem, you will calculate the various probabilities (odds) associated with the game and the expected value of the game.  
    
    In order to calculate the probabilities and expected values - you need some data.   The rules of the game are simple:  Select any 6 numbers of the numbers 1 through 49.  On a ticket, you get to do this twice -- there are two panels containing the numbers 1 through 49.  When you purchase a ticket for $1, you get two panels.   All the information below is from the back of a regular Lotto ticket.
    
    Explain (a) - (f).
    (a).    The odds of getting 6 winning numbers in a panel (1st prize) = 1:13,983,816
    (b).    The odds of getting 6 winning numbers per $1 wagered = 1:6,991,908
    (c).    The odds of getting 5 winning numbers in a panel (2nd prize) = 1:54,201
    (d).    The odds of getting 4 winning numbers in a panel (3rd prize) = 1:1033
    (e).    The odds of getting 3 winning numbers in a panel (4th prize) = 1:57
    (f).     The odds of winning a prize for each $1 played = 1:27
    
    Consider also the following questions:
    (g).    Suppose the jackpot is $6 million.  For simplicity, assume that only first prize winners collect from that pot and that only one person will win the prize.  What is the expected value of a $1 game?  How large would the jackpot have to be for this to be a "fair" game?  A "fair" game is one in which the cost of playing is equal to the expected value for winning.
    (h).    Discuss how these values change if 2 winners split the jackpot. More generally, if multiple winners split the jackpot?
    (i).     What is the expected value of a $1 game?  Assume that first prize is $1,000,000 (i.e., there was a winner last time), 2nd prize is 2.3% of the total lottery sales, 3rd prize is 4.6% of the lottery sales, and 4th prize is a fixed amount of $3. 
    
    Hint:  This situation can be characterized using one of the 6 "standard" discrete distributions.  When you have identified the correct distribution, the calculations should become straightforward.

Questions and Answers

Question 1

On question 4.113. What does it mean to assume binomial approximation to the hypergeometric distribution?

Answer 1

Take a look at pages 132 and 133. This is explained there.

Solutions

Solutions.