# econometrics homework

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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