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
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
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 p
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 pri
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/c
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
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 boo
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 e
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 u
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 di
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
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. Suppos
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 us
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)
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/g
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
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 sampl
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 calc
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
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 be
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 likeli
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
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
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