36-226 Summer 2010
Homework 7
Due July 26
1. Now we will prove that the CDF of any random variable is uniformly distributed on the
interval [0, 1]. We will do this in steps. Let X be a random variable with cdf FX (x) and
inverse cdf FX 1 (x).
(a) Let Y =
36-226 Summer 2010
Homework 8
Solutions
1. (a) For the candidate to have more than 18500 valid signatures, it is necessary that a fraction
18500
of 19850 = 0.932 of the signatures are valid. Thus, we set p = true proportion of
signatures on the petition t
36-226 Summer 2010
Homework 9
Due Aug. 4
START EARLY SO THAT YOU CAN STUDY FOR THE FINAL
1. Let X1 , . . . , Xn exp(). Show that the Inv-Gamma(, ) is conjugate for exponential and
nd the form of the posterior. Use the following forms for the Exponential a
36-226 Summer 2010
Homework 9
Solutions
1.
n
p( | X1 , . . . , Xn )
i=1
1 xi /
e
()
1
1
n exp
1
n+1
+1
1
e/
n
1 /
e
+1
xi
i=1
1
exp (
n
xi + )
i=1
n
Inv-Gamma(n + ,
xi + )
i=1
Thus, since the prior and posterior both come from the Inv-Gamma family, I
36-226 INTRODUCTION TO PROBABILITY & STATISTICS II Midterm II
You must show your work and/or explain your steps in order to get full credit or be considered
for partial credit.
This is a closed book/closed notes exam. You may use a onesided sheet of notes
36-226 INTRO TO PROBABILITY & STATISTICS II N08
PRACTICE FINAL EXAM
You must show your work and/or explain your steps in order to get full credit or be considered
for partial credit.
This is a closed book/closed notes exam. You may use a onesided sheet of
36-226 INTRO TO PROBABILITY & STATISTICS II N08
PRACTICE MIDTERM
You must show your work and/or explain your steps in order to get full credit or be considered
for partial credit.
This is a closed book/closed notes exam. You may use a onesided sheet of no
36-226 Summer 2010
Quiz 1
July 2
1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement.
(a) All estimators are statistics.
TRUE
(b) If is a method of mo
36-226 Summer 2010
Quiz 2
July 9
1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement. You may use
the result above.
(a) The likelihood is always an in
36-226 Summer 2010
Quiz 3
July 23
1. (2 points) What is a pivotal quantity?
A pivotal quantity satises the following two points:
It is a function of the data and the unknown parameter(s)
It has a distribution which does not depend on any unknown paramet
36-226 Summer 2010
Quiz 4
July 30
1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement.
(a) If I fail to reject the null hypothesis with = 0.01 then I
36-226 Summer 2010
Homework 8
Due July 28
1. To get their names on the ballot, political candidates must often produce petitions bearing
the signatures of a minimum number of registered voters. Suppose a specic municipality
requires 18,500 such signatures
36-226 Summer 2010
Homework 7
Solutions
1. (a)
FY (y ) = P(Y < y ) = P(FX (X ) < y ) = P(X < FX 1 (y )
= FX (FX 1 (y ) = y
But this is the CDF of a U(0, 1) random variable, so Y U(0, 1).
(b) We can use the CDF of X , FX (x | ) = 1 ex/ as a pivotal quantit
36-226 Summer 2010
Homework 1
Due June 30
1. Math review.
(a)
n
ln
i=1
ea bxi
xi !
n
=
ln
i=1
n
ea bxi
xi !
xi
a + xi ln b
=
i=1
ln j
j =1
n
= na + ln b
n
xi
xi
i=1
ln j
i=1 j =1
(b)
f (x | a, n) = a ln x + (n a) ln(1 x)
a na
f (x | a, n)
=
0
x
x 1x
a
36-226 Summer 2010
Homework 2
Due July 6
1. Descriptive statistics.
For this problem you will need the data le gdpdata.Rdata available on the website. To
load the data le, use load(/path/to/datafile/gdpdata.Rdata). Once loaded, say
attach(data). It contai
36-226 Summer 2010
Homework 2
Solutions
1. The code for this problem is given at the end.
(a) The histogram is shown in Figure 1 below. The summary statistics are
Min.
0.1
1st Qu.
6.0
Median
24.2
Mean
321.8
3rd Qu.
168.2
Max.
14260.0
(b) This distribution
36-226 Summer 2010
Homework 3
Due July 8
1. Let X1 , . . . , Xn be iid U(0, 2). Find the maximum likelihood estimator (MLE) for , M LE .
Then nd E[M LE ].
2. Let X1 , X2 , . . . , Xn be a random sample from the following distribution:
fX (x) =
I
(x)
x+1 [
36-226 Summer 2010
Homework 3
Solutions
1. X1 , . . . , Xn are iid U (0, 2). Then
1
I
(x)
2 [0,2]
1
I
(x1 , . . . , xn )
L() =
(2)n [0,2]
fX (x) =
Note that you can not nd the mle just by direct calculation in this case (which you can
usually realize if
36-226 Summer 2010
Homework 4
Due July 12
1. Assume that X , the proportion of defective products that a machine produces in a day, has
the following density:
fX (x | ) = (1 x)1 I(0,1) (x).
Note that this is a Beta(1,) distribution.
In order to estimate ,
36-226 Summer 2010
Homework 4
Solutions
1
.
1+
1. (a) Note that this is a Beta(1, ) distribution. E[X ] =
We therefore solve:
1
=
E[X ] = X
= =
=X
1+
1X
X
(b) Compute the mle using calculus as done before.
1
n
L() =
n
(1 xi )
i=1
n
ln L() = n ln + ( 1)
36-226 Summer 2010
Homework 5
Due July 14
1. Let X1 , . . . , Xn be independent but NOT identically distributed. Let Xi Normal(i , 1)
(that is each Xi has a dierent mean i ).
(a) Write down the likelihood and loglikelihood.
(b) Find the MLE for the vector
36-226 Summer 2010
Homework 5
Solutions
1. (a)
Xi N (i , 1)
(independent)
n
L(1 , . . . , n ) =
1
1
2
e 2 (Xi i )
2
i=1
n
2
i=1 (Xi i )
1
n
= (2 ) 2 e 2
n
1
log L(1 , . . . , n ) = log 2
2
2
n
(Xi i )2
i=1
(b) To nd the MLE for each , dierentiate with r
36-226 Summer 2010
Homework 6
Due July 21
1. Suppose that X is distributed N(0, 2 ).
(a) What is the distribution of X 2 / 2 ?
(b) Using the pivotal quantity from part (a), nd a 95% condence interval for 2 of the form a <
2 < b.
(c) Find another 95% CI f
36-226 Summer 2010
Homework 1
Due June 30
1. Math review.
(a) Simplify the following expression as much as possible. The xi are positive integers. There
should be no
signs in the result but there will be
signs.
n
ln
i=1
ea bxi
xi !
.
(1)
(b) Find the valu