Problem Set #1 Economics 43

Due Friday, September 19^th at the start of class Prof. Stephen Schmidt

1. The PDF for rolling two dice and adding the results together is shown below. The odds of a 2 or 12 are 1 in 36. The odds of a 3 or 11 are 2 in 36, of 4 and 10 are 3 in 36, and so forth up to the chance of a 7, which is 6 in 36.
a) What is the chance of rolling either a 6, a 7, or an 8?
b) What is the chance of rolling a number greater than 9? Less than or equal to 5?
c) Find the mean value, variance, and standard deviation of this random variable.
d) Find and graph the cumulative distribution function of this random variable.

2. The graph below shows the probability density function for a random variable Z:
a) What are the largest and smallest values that Z can take?
b) The height of the left side of the PDF is 1/3. What must the height of the right side of the PDF be? (Hint: what must the total area be equal to?)
c) What is the probability that Z is greater than 1? What is the probability that Z is less than 0.5? What is the probability that Z is greater than 2.5?
d) Is the expected value of Z greater than 1, less than 1, or equal to 1? Briefly explain your answer.

3. Suppose the price of oil is distributed normally with a mean of $20 per barrel and a standard deviation of $5.

a) Using the normal distribution table, find the chance that the price will be less than $16 per barrel.
b) Find the chance that the price will be between $22 and $24 per barrel.
c) Is it more likely that the price will be between $22 and $24 per barrel, or between $12 and $14 per barrel? (Do not do any math to answer this question; think about the shape of the normal PDF.)

_4. In this problem we’ll generate some random variables using Excel, and examine their properties using Excel’s built-in statistical tools.

a) Open Excel, and in an empty worksheet, type =rand() in cell A1. Excel will give you a decimal number between 0 and 1, which is a random value drawn from a uniform distribution. Then select cell A1, copy it, select cells A2 to A20, and paste. This will give you 20 random numbers. Note that the value in cell A1 will change when you do the paste, or whenever you change something in the worksheet. To avoid this, select cells A1 to A20, copy them, and select cells B1 to B20. Then go to the Edit menu, select Paste Special, select “Values”, and press OK. Column B will now contain uniform random numbers whose values will not change as you do the rest of this problem. Once you have done this, print the worksheet and turn it in with your answers.
b) How many of the numbers in column B would you expect to be greater than 0.5? How many of them actually are? How many should be greater than 0.9, and how many actually are? How many should be less than 0.1, and how many actually are?
c) In cell B21, type =average(b1:b20) to calculate the average value of the 20 random numbers you have created. What is the expected value of this average? (Hint: use the Law of Large Numbers.) How close is the actual average to its expected value? In cell B22, type =stdev(b1:b20) to calculate the standard deviation of the 20 numbers you have created. The true standard deviation for the uniform distribution is 0.288. Is the standard deviation of your sample close to that?
d) Type something (anything) in cell C1. When you do, the random numbers in column A will change. Using Paste Special, copy the new values into column C of the worksheet. How many of these numbers are greater than 0.5? How many are greater than 0.9, and how many are less than 0.1?
e) Excel can also produce normally distributed random variables. In cell E1, type =norminv(rand(),0,1). This will produce a random variable from the normal density function with mean 0 and standard deviation of 1. Copy cell E1 into cells E2-E20, then use Paste Special to copy the values into column F, where they won’t change as you do the rest of the problem.
f) How many of the values in column F would you expect to be between –1 and 1? How many actually are? How many would you expect to be greater than 2 or less than –2, and how many actually are? Type something in cell G1, then repeat part e) and get a second set of normally distributed random numbers. How many are greater than 2 or less than –2 this time?

_5. In this problem we’ll use Eviews to work with some real data on the income distribution. From the perspective of an econometrician, if we select a person at random from the US population, then that person’s income is random – we don’t know what value it will take, but it could take any value in a range from 0 up to the income of the richest person in the US, and we can find a probability for it falling in any smaller range. (Actually, some people who lose more money in the stock market than they earn in a year can have negative incomes – but to have a negative income you have to be very wealthy, so that you can lose that much in the market.)

You can open Eviews by double-clicking on an Eviews data set. The data set for this problem is called incomes.wf1. The data file is in the Courses server on the Union network. Most computers on campus will have a Courses folder on the desktop; if not, double-click on “Network Neighborhood”, then “Entire Network”, then “Union_acad”, then “Acad_users”, then “course” to get to it. From either place, click on the Economics folder, then on the Datasets folder, then the Eco043 folder, then open the file called pset1.wf1. Or you can open the Eviews program (it should be in the Programs menu on the start button), select the “Open… Workfile” command on the File menu, and navigate to the dataset file. If you wish, you can copy the dataset onto a floppy or Zip disk, which you can then take elsewhere (it is 328K in size). The data set contains observations on 6702 US households taken from the American Housing Survey. It has five variables:

income is the income of the household in dollars
education is the education level of the head of household in years
hispanic is equal to 1 if the household is Hispanic and 0 if not
race is equal to 1 if the household is white, 2 if African-American, 3 if Asian,
and 4 if Native American
urban is equal to 1 if the household is in an urban area and 0 if not

There are also two variables, c and resid, which are in every Eviews worksheet, that we’ll discuss later in the course. Ignore them for now.

a) In the Eviews worksheet, double-click on the urban variable to open it, then click the View button, select Descriptive Statistics, then select Histogram and Stats. What is the mean of this variable? What fraction of households in the sample is located in an urban area? What fraction is in a rural area?
b) Double-click on the income variable to open it, then display its histogram and stats. What is the mean value of income? What is the standard deviation of income? c) Close the income variable. In the worksheet window (the one listing all the variables, click on the Sample button. In the window labeled “If condition”, type “race=2” and click OK. This will cause Eviews to use only the sample that is African-American in calculations. Open the income variable again, and display its histogram and statistics. What is the mean income for African-American households? Is it above or below the average for all households? Is the difference small or large? What explanation(s) might you give for this finding? What is the standard deviation for African-American households? Do African-American households have a wider spread of income distribution, or a narrower range, than all households?
d) Close income, click on the Sample button, and change the If condition to “education>12” to do the calculations for Asians. Display the histogram and stats of income again. What is the average income of households with at least some college education? Is it above or below the average for the population as a whole? How much higher or lower is it? Does this seem like a large amount or a small amount? Is this what you would expect? Why or why not?