ECON 335 – Homework 1 – 100 points
Basic Statistics
DO
NOT WAIT UNTIL THE NIGHT BEFORE THE DUE DATE TO BEGIN THIS HOMEWORK
ASSIGNMENT! IF YOU HAVE TROUBLES DON’T
HESITATE TO EMAIL ME OR COME TO MY OFFICE HOURS WITH QUESTIONS.
This
exercise is designed to give you practice with the concepts of probability
distributions. It contains analytical
exercises that don’t require computer.
Pay close attention to being able to interpret any results you present,
and practice expressing the results in a meaningful way.
ALWAYS
SHOW ALL OF YOUR WORK! A COMPLETE
CORRECT ANSWER SHOULD SHOW ALL WORK LEADING UP TO THE CORRECT ANSWER –
INCLUDING THE EQUATIONS USED. IF YOU
DON’T SHOW ME HOW YOU ARRIVED AT A CALCULATION, I WILL ALSO BE UNABLE TO ASSIGN
ANY PARTIAL CREDIT.
These
homework assignments are designed to complement the material we cover in class,
and give you some practice dealing with the concepts we cover. Think of them as the single best way to
prepare for the exams … putting in more work now means less trouble later. Of course, doing the problems from the book,
or ones you create yourself, etc. will give you even more practice.
_____________________________________________________________________________
1. [25 points] Consider the following
discrete probability distribution function for the variable X:
|
x |
10 |
12 |
14 |
16 |
18 |
|
f(x) |
.02 |
.7 |
.05 |
.2 |
.03 |
a)
[4
pts]
Is this a proper probability mass function? How do you know?
b)
[15
pts]Find the mean, variance, and standard deviation of
X, showing your work.
c)
[6
pts]What is the probability that any draw from this
distribution, say X, is less than 16 [i.e. find P(X<16)]?
2. [40 points]Consider
the following discrete probability mass function for the variables Health Insuranceand Income.
Health Insurance indicates
whether each individual possesses health insurance coverage and, if so, what
type, with
and Income measures gross income class, with
The joint pmf is given
as:
|
HI |
Inc |
||
|
1 |
2 |
3 |
|
|
1 |
.03 |
.13 |
.44 |
|
2 |
.01 |
.01 |
.03 |
|
3 |
.07 |
.03 |
.08 |
|
4 |
.10 |
.06 |
.01 |
a) [4 pts]
Is this a proper probability mass function?
How do you know?
b) [6 pts]
What is the probability that an individual drawn from this distribution holds
public health insurance? What type of probability is this (e.g. joint,
marginal, or conditional)?
c) [9 pts]
Construct the conditional probability distribution (pmf) of Income given that an individual is
covered by a private, employer sponsored health insurance plan; i.e., construct
the table or graph that shows,
.
d) [6 pts]
What is the probability that an individual earns $30,000 or more per year given
that they are insured by at least some form of insurance (i.e. not uninsured)?
e) [6 pts: 2 pts each mean, 2 pts
compare] Calculate
the mean income class for those individuals with private, employer sponsored
insurance. Is this greater than or less
than the mean income class for those with private, individually purchased
insurance? Any thoughts regarding the comparison? [HINT: Use the correct distribution!]
f) [5 pts]
Find the covariance between HI and Inc and provide an interpretation of the
sign of the statistic.
g) [4 pts]
Are HI and Inc independent? How do you
know?
3.
[20 points]The following table give
joint probabilities relating cell phone usage to stopping properly at
intersections. SHOW YOUR WORK!
a)
[5 points]Compute the probability
of failing to stop at an intersection, given the driver was on the cell phone.
b)
[5 points]Compute the probability
of failing to stop at an intersection, given the driver was not using a cell
phone.
c)
[5 points]Compute the probability
of stopping properly at an intersection, given the driver was on a cell phone.
d.
[5 points] Are cell phone usage and
failing to stop at intersections independent of each other? Why or why not?
4.
[15 points]Suppose you are told
that E(X) = 8 and var(X) = 4.
What are the expected values and variances of the following
expressions? HINT: Use the properties of
expected value and variance discussed in Appendix B.
a)
[5 points]
b)
[5 points]
c)
[5 points]
(Hint: you will not be able to solve
explicitly for the variance, but simplify as much as you can).
