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Economics 502-Sections M1 & M2

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Homework 4

Due on Tuesday October 16 at the beginning of the class

This homework has three parts I, II &III. The first part you have to use Excel. In the second part, you have to solve some problems from textbook and outside textbook. In the third part, you solve some problems assigned from the textbook. You are required to do parts I & II. Part III is optional. I. Use Excel for the following: You will turn in the print of your completed Excel worksheets. i) First sheet Suppose a population contains a finite number N of elements that possess one of two characteristics. Thus, r of the elements might be of one type (for example type A) and N-r are of another type (for example type B). A sample of n elements is randomly selected from the population, and the random variable of interest is Y, the number of type A elements in the

sample. Then Y has a hypergeometric distribution and we know: 𝑃(𝑌 = 𝑦) = ( 𝑟 𝑦)(

𝑁−𝑟 𝑛−𝑦)

(𝑁𝑛)

Now let denote by p, the probability of a randomly selected element is of type A. Then, p ≈ r/N. Let’s assume as N increases, r will also change in the fixed proportion to it in such a way to keep p fixed (i.e. r = N*p where p is fixed). We would like to show that the hypergeometric approaches the binomial, i.e.

lim 𝑁→∞ 𝑟=𝑁𝑝 𝑝 𝑓𝑖𝑥𝑒𝑑

( 𝑟 𝑦) (

𝑁 − 𝑟 𝑛 − 𝑦

)

( 𝑁 𝑛 )

= ( 𝑛 𝑦) 𝑝

𝑦(1 − 𝑝)𝑛−𝑦

To show this, follow the instructions below in Excel: A1 Name A2 UIN Enter the following in the cells C1 to I1:

n y p Binomial N r HyperGeometric Enter in cell: C2: the last two digit of your UIN D2: 1/3 of the cell 2. Round it to the closest whole number. E2 enter 0.6 F2: Calculate the binomial probability based on the values n, y and p. G2: enter 1000 H2: \$E\$2* G2 I2: Calculate the hypergeometric probability based on the values n, y, N and r. G3: G2 + 1000 Copy & paste cells G3:I3 down. Does the hypergeometric approach the binomial?

ii) Second sheet We have 5 people drawn from a population in which p = 0.ab are smokers, where ab are the last two digits of your UIN. We want to tabulate the probability distribution function and the cumulative distribution function of the number of smokers X in our group of n = 5 people. In Excel do the following: A1 Name A2 Section A3 UIN A4 n B4 5 A5 p B5 0.ab Make a table as follows: In the first row put the following: D6 X E6 P(X=x) F6 F(X=x), where P(X=x) gives you the probability and F(X=x) is the cumulative distribution function. Put all the possible values of X in column D under X and then calculate P(X=x) and F(X=x) and complete the table. Draw the bar charts for P(X=x) and F(X=x). II. Solve the following problems 1) Suppose the range of a discrete random variable is {0, 1, 2, 3, 4} and its probability mass function is f (x) = x/10 . What is its cumulative distribution function? 2) Roll a die twice and record the outcomes as (i, j), where i is the result of the first roll and j the result of the second. Define the random variable M to be the maximum value of the two dice: M(i, j) = max(i, j). a) Write down and graph the probability mass function of M, P(M=m). b) Write down and graph the cumulative distribution function of M, F(M=m). c) What is F(4), F(4+) and F(4-)? 3) A discrete random variable X has the cumulative distribution function

𝐹(𝑥) =

{

0 𝑥 < 0 0.1 0 ≤ 𝑥 < 1 0.3 1 ≤ 𝑥 < 2 0.5 2 ≤ 𝑥 < 4 0.8 4 ≤ 𝑥 < 5 1 5 ≤ 𝑥

a) Determine the probability mass function of X. b) What is F(1+) and F(1-)? 4) The probability density function if the failure time of a pump is uniform (i.e. it has a constant value, k) between 0 and 100 months and zero elsewhere. a) Find k b) Find and plot the hazard function, h(t). c) Find the expected life of the pump.

5) Do the following problems from textbook. In some of these problems, you have to use APPLETS (a short computer application especially for performing a simple specific task). You can access the APPLETS in the following site:

Chapter 3: 7th edition: 170 6th edition: 134 Chapter 4: 7th edition: 8, 10, 28, 48, 66, 72, 84, 94, 96, 140, 144, 146, 169, 187 6th edition: 4, ??, 22, 36, 52, 58, ??, 74, 76, 108, 112, 136, 137, 153 III. (Optional) Solve the following problems from the textbook Chapter 3: 7th edition: 167, 6th edition: 131, Chapter 4: 7th edition: 16, 20, 56, 64, 106, 168, 6th edition: 10, 14, 44, 50, 84, 136,

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