ECON 3303 SMU Probability Distribution Economic Statistics Questions

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ECON 3303/5503
Assignment 1
Atul A. Dar
September 30, 2021
Submission deadline: You must upload your handwritten answers in a single PDF file to the Assignments folder in
Brightspace by 9 pm on October 9. I will post solutions at 9:15pm and cannot, therefore, accept any submission after
that time. Please keep in mind that, in order to be fair to all students, I cannot accept any special requests for an
extension. Also remember that the assignment can be completed well within the allowed time frame, and I would
urge you not to leave it to the last moment.
The assignment can be done in groups of 2 or 3 students, but individual submissions are welcome.
Answer the following questions. Make sure you show how you arrived at your answers.
Question 1 (10 points) Parts A and B are independent
A. Prove each of the following using the properties of the expectations or summation operator. (6 points)
a. E(X-?)2 = E(X2)- ?2
c. If n=3, (?xi)2 = ?xi2 + 2(x1x2 + x1x3 +x2x3).
b. E[(X-?X )(Y- ?Y)] = E(XY) – ?X?Y
d. ?(bxi + cyi)2= b2(?xi2 +?yi2 + 2?xiyi) when b=c.
Bonus point: ?(xi – a)2 =0 given ?(xi – a)=0.
B. Suppose the probability distribution of random variable X is as follows.
x:
f(x):
a
0.4
2a
0.2
3a
b
4a
0.5b
5a
0.04
Note that a and b are some unknown numbers. Suppose the mean of X, E(X) = 11.
i. Determine the values of a and b from the information provided. (2 points)
ii. Using the solutions for a and b, obtain the numerical value of E(X2). (2 points)
Question 2 (6 points)
The table below gives the joint probability distribution of two random variables X and Y.
X
Y
10
20
30
f(x)
0
0.125
1
0.0
??
??
??
0.125
0.125
??
2
0.0
0.25
0.125
0.375
3
0.125
0.0
0.0
0.125
f(y)
0.25
??
0.25
??
The means of X and Y are: ?X =1.5 and ?Y = 20.
1.
2.
3.
4.
Complete the joint pdf entries indicated by ??. (2 points)
Are X and Y statistically independent? Explain. (1 point)
Are X and Y linearly related? Explain. (2 points)
Can you explain the differences (if any) in your answers to 2. and 3.? (1 point)
Question 3 (6 points)
Suppose the total income of individuals is made up of three random components: income from work X (wages), income
from assets Y, and government transfers T (such as pensions, welfare payments, etc). The government taxes non-asset
income (that is, wage income and transfer income) at the same rate t, while it taxes asset income at the rate s. Note:
these tax rates are constants and are in decimals. For example, a 10 percent tax rate on asset income means that s=0.1;
t is to be interpreted in the same way.
a. Write down the expression for total income after tax. Denote this as Z. (1 point)
b. Find an expression for E(Z). Then find an expression for var(Z), assuming that cov(X, Y)= cov(T, Y)= 0 but cov (X, T)=10.
Simplify the variance expression as much as you can. (3 points)
c. How would your answers to a. and b. change if t=s=0 and there was a flat (that is, constant) tax of $h on nonasset income and $k on asset income? Note: both h and k are constants. Show your work fully. (2 points)
Question 4 (8 points)
According to theory, in the joint probability distribution of a household’s expenditure on alcoholic beverages (Y) and
its income (X), the former, on average, is expected to be positively related to income. If this relationship is linear, we can
write it as: E(Y|x)= ?1 +?2x. You are to examine this relationship in the context of Canadian households.
Suppose you are given the following information on the joint pdf of Y and X for Canadian households, where each
variable is in thousands of dollars, and represents annual values. The data are taken from the 1992 Family Expenditure
survey.
?xy= 0.3053, ?X2 =321.990, ?Y2=3.499, 3.6572, ?X = 33.132 and ?Y = 1.643.
Note: although this information is based on a survey, we can assume that it represents the underlying population for the
analysis required in this question.
In your computations below, you can round off decimals to three spaces.
a. Use this information to obtain and interpret the meaning of ?1 and ?2. (3 points)
b. Consider households with an income of $50 thousand. By how much would average expenditure on alcoholic
beverages be higher for households with an income that is (i) higher by $10 thousand, and (ii) higher by 10%?
Show your work. You must calculate the precise difference (3 points)
c. Suppose that the opening of a new store was being contemplated in some community. The opening would be
economically viable only if average annual household expenditure on alcoholic was at least $1.5 thousand in that
community. Calculate the minimum income required to make the store economically viable. Use your equation
to answer this. (2 points)

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Tags:
minimum income

statistical independence

capital sigma

relationship between x and Y

distribution of household expenditure

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