Problem set 1 Assignment Rules 1. Homework assignments must be typed. For instruction how to type equations and math objects please see notes “Typing Math in MS Word”. 2. 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. 3. Your assignments must be stapled. 4. 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. 5. Homework assignments must be submitted at the end of the lecture, in class, on the listed dates. 6. Late homework assignments will not be accepted under any circumstances, but the lowest homework score will be dropped. 7. The first homework assignment cannot be dropped. 8. All the graphs should be fully labeled, i.e. with a title, labeled axis and labeled curves. 9. 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. 10. This page must be part of the submitted homework. 1. Suppose that length of life in Japan, X , has exponential distribution: ) ( ~ b EXP X . The pdf of X is given by: otherwise 0 0 , ) ( ³ î í ì = - x e x f x b b a. What is the support of X ? b. Prove that indeed, the above function is a pdf (i.e. nonnegative on the entire support, and integrates to 1 over the entire support). c. Show that life expectancy in Japan is b 1 ) ( = X E . (Hint: use integration by parts). d. Show that the probability that a newborn will live until the age of 100 is b 100 - e . e. Suppose that only 5% of the newborns live more than the age of * x . Show that b - = 05 . 0 ln * x . 2. Consider the random experiment of tossing two dice. a. Write the sample space for this random experiment. b. 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). c. Show the probability density function of X. The best way to do this is to create a table like this: x ) ( x f 1 36 1 2 36 3 … … d. Calculate the expected value (mean) of X. e. Calculate the variance of X. 3. Let X be a continuous random variable, with pdf î í ì £ £ - = otherwise 0 2 0 5 . 0 1 ) ( x x x f a. Verify that f is indeed a probability density function (i.e. it is nonnegative, and integrates to 1 over the entire support). b. Using Excel, plot the graph of this pdf. c. Calculate the mean of X. d. Calculate the variance of X. 4. Let X be a random variable with mean m and variance 2 s , and let s m - = X Y . a. Using rules of expected values show that the mean of Y is 0. b. Using the rules of variances, show that the variance of Y is 1. 5. Consider the function î í ì £ £ £ £ - - = otherwise 0 1 0 ; 1 0 2 ) , ( y x y x y x f a. Show that ) , ( y x f is a probability density function ...

1. Problem set 1
Assignment Rules
1. Homework assignments must be typed. For instruction how to
type equations and math objects please see notes “Typing Math
in MS Word”.
2. 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.
3. Your assignments must be stapled.
4. 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.
5. Homework assignments must be submitted at the end of the
lecture, in class, on the listed dates.
6. Late homework assignments will not be accepted under any
circumstances, but the lowest homework score will be dropped.
7. The first homework assignment cannot be dropped.
8. All the graphs should be fully labeled, i.e. with a title,
labeled axis and labeled curves.
9. 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.
10. This page must be part of the submitted homework.
1. Suppose that length of life in Japan,
X

2. , has exponential distribution:
)
(
~
b
EXP
X
. The pdf of X is given by:
otherwise
0
0
,
)
(
³
î
í
ì
=
-
x
e
x
f
x
b
b
a. What is the support of

3. X
?
b. Prove that indeed, the above function is a pdf (i.e.
nonnegative on the entire support, and integrates to 1 over the
entire support).
c. Show that life expectancy in Japan is
b
1
)
(
=
X
E
. (Hint: use integration by parts).
d. Show that the probability that a newborn will live until the
age of 100 is
b
100
-
e
.
e. Suppose that only 5% of the newborns live more than the age
of

4. *
x
. Show that
b
-
=
05
.
0
ln
*
x
.
2. Consider the random experiment of tossing two dice.
a. Write the sample space for this random experiment.
b. 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).
c. Show the probability density function of X. The best way to
do this is to create a table like this:
x
)
(
x

5. f
1
36
1
2
36
3
…
…
d. Calculate the expected value (mean) of X.
e. Calculate the variance of X.
3. Let X be a continuous random variable, with pdf
î
í
ì
£
£
-
=
otherwise
0
2
0
5
.

6. 0
1
)
(
x
x
x
f
a. Verify that f is indeed a probability density function (i.e. it is
nonnegative, and integrates to 1 over the entire support).
b. Using Excel, plot the graph of this pdf.
c. Calculate the mean of X.
d. Calculate the variance of X.
4. Let X be a random variable with mean
m
and variance
2
s
, and let
s
m
-
=
X
Y
.

7. a. Using rules of expected values show that the mean of Y is 0.
b. Using the rules of variances, show that the variance of Y is 1.
5. Consider the function
î
í
ì
£
£
£
£
-
-
=
otherwise
0
1
0
;
1
0
2
)
,
(
y
x
y
x
y
x
f

8. a. Show that
)
,
(
y
x
f
is a probability density function.
b. Check whether
X
and
Y
are statistically independent.
6. Let X be a random variables, and a, b be some numbers. Let
b
aX
Y
+
=
. Prove that: if
0
>
a

9. , then
1
)
,
(
=
Y
X
corr
, if
0
<
a
then
1
)
,
(
-
=
Y
X
corr
, and if
0
=
a

10. , then
0
)
,
(
=
Y
X
corr
.
7. Meteorologists study the correlation between humidity H, and
temperature. Some measure the temperature in Fahrenheit F,
while others use Celsius C, where
(
)
32
9
5
-
=
F
C
.
a. 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

11. (
)
(
)
C
H
corr
F
H
corr
,
,
=
b. Will the researchers get the same covariance if they use
different units? Prove your answer.
c. Based on your answers to a and b, should researchers report
covariance or correlation from their studies? Why?
8. Let
1
X
and
2
X
be identically distributed random variables, and thus both have
the same mean
m
and variance

12. 2
s
. Let
X
be the average of
1
X
and
2
X
, that is
2
1
2
1
2
1
X
X
X
+
=
.
a. Show that the mean of
X

13. is
m
.
b. Find the variance of
X
.
c. Show that if
1
X
and
2
X
are independent, then the variance of
X
is
2
2
s
.
9. This question generalizes the previous one to average of any

14. number of identically distributed random variables. Let
n
X
X
,...,
1
be n identically distributed random variables with mean
m
and variance
2
s
. Let the average of these variables be
å
=
=
n
i
i
n
X
n
X
1
1
.
a. Show that the mean of

15. n
X
is
m
.
b. Show that if
n
X
X
,...,
1
are independent, then the variance of
n
X
is
n
2
s
.
c. What is the limit of
(
)
n

16. X
var
as
¥
®
n
, still assuming that
n
X
X
,...,
1
are independent?
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