PS04 - University of Illinois Fall 2009 ECE 313: Problem...

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University of Illinois Fall 2009 ECE 313: Problem Set 4 Counting Random Variables, Maximum-Likelihood Estimation Due: Wednesday September 23 at 4 p.m. . Reading: Ross, Chapter 4 Noncredit Exercises: DO NOT turn these in. Chapter 4: Problems 32, 38-39, 40-43, 47-52; Theoretical Exercises 16-19; Self-Test Problems 9, 13, 15, 16. 1. [Graphical study of binomial pmfs] Use a spreadsheet/Mathematica/MATLAB for this problem. Let A denote an event of probability p . (a) For p = 0 . 1 , 0 . 25 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 75, and 0 . 9, find the numerical values of the probabilities that A occurs 0 , 1 , 2 ,..., 10 times on 10 trials. (b) You have computed the pmf of a binomial random variable X p with parameters (10 ,p ) for seven choices of p . For each value of p , draw a bar graph of the pmf of X p . (The pmf of X 0 . 5 is shown on page 157 of the text!). (c) What is the relationship between the pmfs of X p and X 1 - p ? 2. [The probability of an even number of successes] Let X denote a binomial random variable with parameters ( N,p ). What is P { X is even } ? Hint: Expand ( x + y ) n + ( x - y ) n using the binomial theorem and then set x = 1 - p , y = p . 3. [The pricing of airline tickets] Eight persons have purchased tickets ($ F per person) for travel in a 5-passenger plane on a scheduled airline flight in Ruritania. The number of persons who actually show up to travel can be modeled as a binomial random variable X with parameters (8 , 0 . 5). Naturally, if more than 5 persons show up, only
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PS04 - University of Illinois Fall 2009 ECE 313: Problem...

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