SOLUTION: University of Houston Probability Theory Expected Value and Lotteries Questions

Econometrics PS 1
Due: Feb 2
Complete the entire problems in each section as required. There are two sections!
Section 1: Probability theory: Expected Value and Lotteries (ONE QUESTION TOTAL)
We discussed how the sample mean can be skewed by an extreme value. In a sample of 100 people
from Texas, if a multi-millionaire oil baron is randomly chosen for the sample, the mean income in the
sample would be skewed higher than the median.
The sample mean is sometimes referred to as the expected value, written E[X] for the expected value of
X. In probability theory the expected value is the sum of all potential outcomes, weighted by the
probability/chance of the occurrence. For example, assume you are close friends with the oil baron. You
need money for school, and as your friend he will agrees to give you one of two cars he never drives,
which you will immediately sell for cash. He will flip a coin to determine which one, giving you a 50/50
chance of each. Let’s say one car is worth $12,000 and the other car is worth $118,000. The expected
value is written E[X] = 0.5 × $12,000 + 0.5 × $118,000 = $65,000. Notice that this is also the average of
the two values.
Question 1: Consider a random lottery, where 2,250 people enter their name (only once perperson) and a machine selects one winner at random. Each player has an equal chance of
selection and the winning prize is $425,000.
a. Absent any costs associated with winning or playing the lottery, what is the expected value
of entering the lottery one time?
b. Assume the winner must pay a 20% tax on lottery winnings. Further, the dealer wants to
charge an entry fee. Exactly 2,250 people believe in luck and will play if the expected value
of the gamble is greater than or equal to zero. What is the maximum entry fee the dealer
can charge? Will the dealer make a profit?
c. What is the fundamental difference between the typical Powerball or Megamillions lottery
and the one we established in our example above? Use two sentences or less to explain.
Section 2: Stata Exercises (FIVE QUESTIONS TOTAL)
For this section you will download data from this site:
Connect to the virtual Stata console here:

Find the data “labsup.dta” which contains variables for birthing mothers and their background
characteristics. Save the file to your computer in a folder you can track down.
Boot up StataSE 15 by clicking on the icon .
(File Transfer)

Upload the labsup.dta file to your virtual desktop using file
transfer. Open the tab on the far left side of the console and
select the file transfer option (circled in red in the
screenshot). Your file will be stored in the virtual
“documents” folder. You will also use this tab to toggle
between open files (circled in blue).

Stata allows us to write programs, so our work is reproducible. We save the program codes as
.do file format, which is Stata specific but other statistical software have the same functionality.
Once Stata is up and running, click the icon to open a new .do file. Of the two with a pencil, it is
the one on the left.

Before proceeding, save your .do to the documents folder. In the .do file pane, click “file…save
as…”, select desktop, then your user folder, then documents folder. While you have the window
open, save the filepath by right clicking on “Documents” at the top and selecting “Copy address
as text.”

Navigate back to your .do file. The first step is to set the working directory using the cd (change
directory) command. We will use the documents folder as the directory. Next, load the dataset
with the use command. In this example I clear the temporary memory, set the working
directory, and load the data. Select the block you wish to execute and press the “do” button.
By this point you will see the data loaded up in the main Stata panel. That’s the way it works.
You code up something in the .do file and it is executed in the workstation.
Other than the .do file you can tell Stata to execute something by typing directly into the
command line in the main Stata panel. Note that for any Stata function you can use help
followed by the command name, directly into the command line. Also note that to see the
data table, type browse into the command line and press enter.
1. The labsup dataset contains variables describing the fertility of n=31,857 mothers.
Summarize the variable “kidcount” in detail and answer the following questions.
a. Describe the sample based on the number of kids. Are there any women in the
sample with no children? Use three sentences or less.
b. Let’s say a big family is one that has three kids or more. Based on this definition,
what fraction of the total sample has a big family? Use the count function.
c. If I were to randomly select a mother from this sample, what is the probability that I
would choose a woman with 5 kids or more? What about 3 kids or less?
2. Random variables can be categorized as discrete or continuous. A discrete variable is one
that has values determined by counts: the number of kids, total family size, etc. A
continuous variable is measured and can take on fractional values: income, miles, square
footage of a house are a few examples.
a. Based on your summary output in #1, is the median or mean a better descriptor for
the number of kids each mother has? Why?
b. There are three income variables in the dataset. Use the kdensity function to plot
the probability density for “labinc” (mom’s labor income in 000’s). Also summarize
“labinc.” What is informative from looking at the probability density that you might
not have picked up just by looking at the mean? Paste a screenshot of your density
here as well.
c. Now plot the probability density for “faminc” (total household income). Does this
variable approximate a normal distribution? Are there observations in the data you
might flag as erroneous?
3. Use the correlation command to create a single correlation matrix for the following
variables: agefstm (mother’s age at first birth), kidcount, faminc, and educ (years of
a. Paste a screenshot of your correlation matrix here.
b. Some of the correlations are obvious, for example years of education is positively
correlated with income. Identify one positive and one negative correlation that
would have been less obvious before running this correlation.
c. A researcher looks at the correlation between family income and kidcount and
notices that the sign is negative. She then says to herself, “wow, lower income
families have more kids. This is surprising given they would have less resources than
a richer family.” Think like an economist- given the correlation between family
income, years of education, and mother’s age at first birth, what might be an
alternative explanation for why lower income mothers have more children? Use
four sentences or less.
4. Indeed, the correlation between family income and number of kids is not very
strong. Let’s run a regression to test for the effect of increasing income on the
number of kids. Using the regress command, run the following regression equations.
a. First run the regression
where Yi is kidscount, and X1i is faminc (family income). What is the
coefficient for B1?
b. Now run the regression
Where X2i is years of education and X3i is mom’s age at first birth. Does the
coefficient B1 become smaller or larger?
c. Now run the regression
Where Zi is labinc (mom’s labor income). Which coefficient is larger, the
coefficient for family income (B1) or the coefficient for mom’s income (B4)?
Also paste a screenshot of your regression results for part c (only) here.
5. In the 1980’s the FDA approved oral contraceptives commonly referred to as birth control
pills. Think like an economist – what is the opportunity cost of childbirth for a woman of
working age? What effect do you think birth control pills had on college graduation rates of
women since the approval? Given what you found in the prior steps, what is one mechanism
in which birth control pills might decrease the fertility rate in the United States, outside of
directly preventing contraception? Write two paragraphs or less summarizing your thoughts
about these questions.

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