econometrics homework

| October 22, 2018

San
Francisco State University

Michael
Bar

ECON
312

Fall
2013

Problem set 1
Due Thursday,
September 5, in class

Name
(type)______________________________

Assignment Rules

Homework
assignments must be typed. For instruction how to type equations and math
objects please see notes “Typing Math in MS Word”.
Homework
assignments must be prepared within this template. Save this file on your
computer and type your answers following each question. Do not delete the
questions.
Your
assignments must be stapled.
No
attachments are allowed. This means that all your work must be done within
this word document and attaching graphs, questions or other material is
prohibited.
Homework
assignments must be submitted at the end of the lecture, in class, on the
listed dates.
Late
homework assignments will not be accepted under any circumstances, but the
lowest homework score will be dropped.
The first
homework assignment cannot be dropped.
All the
graphs should be fully labeled, i.e. with a title, labeled axis and labeled
curves.
In all
the questions that involve calculations, you are required to show all your
work. That is, you need to write the steps that you made in order to get
to the solution.
This page
must be part of the submitted homework.

Suppose
that length of life in Japan,.0/msohtmlclip1/01/clip_image002.gif”>, has exponential distribution: .0/msohtmlclip1/01/clip_image004.gif”>. The pdf of X is given by:

.0/msohtmlclip1/01/clip_image006.gif”>

What is the support of.0/msohtmlclip1/01/clip_image002.gif”>?
Prove that indeed, the above function is a pdf (i.e. nonnegative on
the entire support, and integrates to 1 over the entire support).
Show that life expectancy in Japan is .0/msohtmlclip1/01/clip_image009.gif”>. (Hint: use integration by parts).
Show that the probability that a newborn will live until the age of
100 is.0/msohtmlclip1/01/clip_image011.gif”>.
Suppose that only 5% of the newborns live more than the age of.0/msohtmlclip1/01/clip_image013.gif”>. Show that .0/msohtmlclip1/01/clip_image015.gif”>.

Consider
the random experiment of tossing two dice.

Write the sample space for this random experiment.
Let X be a random variable, which records the maximum of the
two dice. List all the possible values of X (i.e., describe the
support of X).
Show the probability density function of X. The best way to
do this is to create a table like this:

.0/msohtmlclip1/01/clip_image017.gif”>

.0/msohtmlclip1/01/clip_image019.gif”>

1

.0/msohtmlclip1/01/clip_image021.gif”>

2

.0/msohtmlclip1/01/clip_image023.gif”>

Calculate the expected value (mean) of X.
Calculate the variance of X.

Let
X be a continuous random variable, with pdf

.0/msohtmlclip1/01/clip_image025.gif”>

Verify that f is indeed a probability density function (i.e.
it is nonnegative, and integrates to 1 over the entire support).
Using Excel, plot the graph of this pdf.
Calculate the mean of X.
Calculate the variance of X.

Let
X be a random variable with mean .0/msohtmlclip1/01/clip_image027.gif”> and variance .0/msohtmlclip1/01/clip_image029.gif”>, and let .0/msohtmlclip1/01/clip_image031.gif”>.

Using rules of expected values show that the mean of Y is 0.
Using the rules of variances, show that the variance of Y is
1.

Consider
the function

.0/msohtmlclip1/01/clip_image033.gif”>

Show that
.0/msohtmlclip1/01/clip_image035.gif”> is a
probability density function.
Check
whether .0/msohtmlclip1/01/clip_image002.gif”> and .0/msohtmlclip1/01/clip_image038.gif”> are
statistically independent.

Let X be a random variables, and a,
b be some numbers. Let .0/msohtmlclip1/01/clip_image040.gif”>. Prove that:if .0/msohtmlclip1/01/clip_image042.gif”>, then .0/msohtmlclip1/01/clip_image044.gif”>, if .0/msohtmlclip1/01/clip_image046.gif”> then .0/msohtmlclip1/01/clip_image048.gif”>, and if .0/msohtmlclip1/01/clip_image050.gif”>, then .0/msohtmlclip1/01/clip_image052.gif”>.

Meteorologists
study the correlation between humidity H, and temperature. Some
measure the temperature in Fahrenheit F, while others use Celsius C,
where .0/msohtmlclip1/01/clip_image054.gif”> .

Show
that two researchers, who use the same data, but measure temperature in
different units, will nevertheless find the same correlation between
humidity and temperature. In other words, show that

.0/msohtmlclip1/01/clip_image056.gif”>

Will the researchers get the same
covariance if they use different units? Prove your answer.
Based on
your answers to a and b, should researchers report covariance or
correlation from their studies? Why?

Let .0/msohtmlclip1/01/clip_image058.gif”> and .0/msohtmlclip1/01/clip_image060.gif”> be
identically distributed random variables, and thus both have the same mean .0/msohtmlclip1/01/clip_image027.gif”> and variance .0/msohtmlclip1/01/clip_image029.gif”>. Let .0/msohtmlclip1/01/clip_image062.gif”> be the average of
.0/msohtmlclip1/01/clip_image058.gif”> and .0/msohtmlclip1/01/clip_image060.gif”>, that is .0/msohtmlclip1/01/clip_image064.gif”> .

Show
that the mean of .0/msohtmlclip1/01/clip_image062.gif”> is .0/msohtmlclip1/01/clip_image027.gif”>.
Find the variance of .0/msohtmlclip1/01/clip_image062.gif”>.
Show that if .0/msohtmlclip1/01/clip_image058.gif”> and .0/msohtmlclip1/01/clip_image060.gif”> are independent, then the variance of .0/msohtmlclip1/01/clip_image062.gif”> is .0/msohtmlclip1/01/clip_image067.gif”>.

This
question generalizes the previous one to average of any number of
identically distributed random variables. Let .0/msohtmlclip1/01/clip_image069.gif”> be n
identically distributed random variables with mean .0/msohtmlclip1/01/clip_image027.gif”> and variance .0/msohtmlclip1/01/clip_image029.gif”>. Let the average of these variables be .0/msohtmlclip1/01/clip_image071.gif”>.

Show
that the mean of .0/msohtmlclip1/01/clip_image073.gif”> is .0/msohtmlclip1/01/clip_image027.gif”>.
Show that if .0/msohtmlclip1/01/clip_image069.gif”> are independent,
then the variance of.0/msohtmlclip1/01/clip_image073.gif”> is .0/msohtmlclip1/01/clip_image075.gif”>.
What is
the limit of .0/msohtmlclip1/01/clip_image077.gif”> as .0/msohtmlclip1/01/clip_image079.gif”>, still assuming that .0/msohtmlclip1/01/clip_image081.gif”> are independent?

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