APPM 4/5520
Solutions to Problem Set Four
1. Let X bin(n, p).
=
n
x=0
bin. thm.
2.
n
x
px (1 pnx )
(pet )x (1 p)nx
(pet + 1 p)n
=
n
x
n
tx
x=0 e
E[etX ] =
=
M (t)
M (t) = E[etX ] =
tx
e
=
tx
0e
=
(t)x
0 e
f (x) dx
ex dx
dx
Note that we need t < or els

APPM/MATH 4/5520
Solutions to Exam II Review Problems
1.
FYn (y ) = P (Yn y ) = P (n ln(X(1) + 1) y )
= P (X(1) ey/n 1)
Since
P (X(1) x)
=
1 P (X(1) > x)
iid
1 [P (X1 > x)]n
=
P areto
=
1
(1+x)
=
we have that
1
1
n
1
(1+x)n ,
FYn (y ) = P (X(1) ey/n 1)
=

APPM/MATH 4/5520
Solutions to Exam I Review Problems
1. First note that P (X = x) = (1 p)x p Icfw_0,1,2,. . So,
P (Y = y ) = P (X + 1 = y ) = P (X = y 1)
= (1 p)y1 p Icfw_0,1,2,., (y 1)
= (1 p)y1 p Icfw_1,2,3,., (y )
Therefore, Y geom1 (p).
2. y = g(x) =

APPM/MATH 4/5520
Solutions to Problem Set Eleven
1. The eciency of S 2 as an unbiased stimator of 2 is dened to be
CRLB2
V ar (S 2 )
Since (n 1)S 2 / 2 2 (n 1), we can easily compute the variance of S 2 :
V ar (S 2 ) = V ar
2 (n 1)S 2
n1
2
=
4
V ar (W )

APPM/MATH 4/5520
Problem Set Twelve (Due Wednesday, December 11th)
This homework is OPTIONAL. As of now, your three lowest homework scores are dropped. If
you do this assignment, you can drop four homework scores!
1. Let X1 , X2 , . . . , Xn be a random s

APPM/MATH 4/5520
Problem Set Eleven (Due Wednesday, December 4th)
1. Let X1 , X2 , . . . , Xn be a random sample from the N (, 2 ) distribution where is known.
Find the eciency of the estimator 2 = S 2 .
2. Let X1 , X2 , . . . , Xn be a random sample from

APPM 4/5520
Solutions to Problem Set Nine
1. (a) The power function for the X(1) test is:
1 () = P ( Reject H0 ; )
= P X(1) < 1n ln(1 );
0
n 1n ln(1)
= 1e
0
= 1 (1 )/0 .
(b) Since large values of are reected by small values in the sample (including a sma

APPM/MATH 4/5520
Problem Set Ten (Due Wednesday, November 13th)
1. Let X1 , X2 , . . . , Xn be a random sample from the (3, ) distribution. Find the MLE (maxi2
mum likelihood estimator) of E[X1 ] based on the entire sample.
2. Let X1 , X2 , . . . , Xn be

APPM/MATH 4/5520
Solutions to Problem Set Eight
1. (a) First note that Y = F (X ) and F a cdf implies that Y takes values in [0, 1]. Now
FY (y ) = P (Y y ) = P (F (X ) y ) = P (X F 1 (y ) = F (F 1 (y ) = y
so
fY (y ) =
d
fY (y ) = 1
dy
for y [0, 1].
There

APPM/MATH 4/5520
Problem Set Nine (Due Wednesday, November 6th)
1. Suppose that you have a random sample, X1 , . . . , Xn from an exponential rate distribution
and that you would like to come up with a test for H0 : 0 versus H1 : > 0 .
(a) On the last hom

APPM/MATH 4/5520
Solutions to Problem Set Seven
1. The appropriate z -critical value is z0.075 = 1.44.
So, the condence interval is
x 1.44
n
3
2.8 1.44
5
(1.68, 3.92)
2. Recall that we dene a t-distribution with n degress of freedom by letting Z N (0, 1

APPM/MATH 4/5520
Problem Set Eight (Due Wednesday, October 30th)
1. Throughout this problem, you may assume that F in invertible.
(a) Let X be a continuous random variable with cdf F . Find the distribution of Y = F (X ).
(b) Show or argue that Y , dened

APPM 4/5520
Problem Set Seven (Due Wednesday, October 23rd)
1. Suppose that a random sample of size 5, taken from the N (, 3) distribution, results in a
sample mean of 2.8. Give an 85% condence interval for the true mean .
2. Let X1 , X2 be a random sampl

APPM/MATH 4/5520
Solutions to Problem Set Five
1. (a) The cdf for X(1) is
FX(1) (x) = P (X(1) x) = 1 P (X(1) > x)
iid
= 1 P (X1 > x)n
= 1 [1 FX1 (x)]n
= 1 1 (1 e1/ x )
n
= 1 en/ x
So, we have that X(1) exp(rate = n/ ). (Alternatively, you could have just

APPM/MATH 4/5520
Problem Set Five (Due Wednesday, October 2nd)
1. Let X1 , X2 , . . . , Xn be a random sample from the exponential distribution with rate .
Consider estimating where is dened as 1/. Note that is the mean of this distribution.
We already kn

APPM/MATH 4/5520
Problem Set Six (Due Wednesday, October 16th)
Note: In what follows, nd the limiting distribution means that it is a convergence in distribution
problem.
iid
1. Suppose that X1 , X2 , . . . , Xn N (, 2 ). In class we proved that
(n 1)S 2

APPM 4/5520
Problem Set Four (Due Wednesday, September 25th)
1. Use the Binomial Theorem to nd the moment generating function for the binomial distribution with parameters n and p.
2. Derive the moment generating function for the exponential distribution

APPM/Math 4/5520
Solutions to Final Exam Review Problems
1. (a)
f (x; ) = n
(1+ )
L( ) = n
(1 + xi )
(1 + xi )
I(0,) (xi )
(1+ )
( ) = ln L( ) = n ln (1 + )
d
n
( ) =
d
set
ln(1 + xi ) = 0
n
ln(1 + Xi )
n =
ln(1 + xi )
(b) MLEs are always asymptotically

APPM 4/5520
Solutions to Problem Set One
1. P (X = x) = P (x failures before the r th success)
To have x failures before the r th success, we will have to observe a total of x + r trials in
which there will be exactly x failures. Since we stop looking whe

APPM 4/5520
Problem Set Three (Due Wednesday, September 18th)
iid
1. Suppose that X1 , X2 exp(rate = ). Find the distribution of
X1
X1 + X2 .
(Name it!)
2. Suppose that X1 , X2 , . . . , Xn is a random sample from the uniform distribution over the
interva

APPM/MATH 4/5520
Solutions to Problem Set One
1. Recall the nite geometric sum
N
n
n=0 r
= (1 r N +1 )/(1 r ).
F (x) = P (X x) = P (X = 0) + P (X = 1) + + P (X = x)
=
x
u=0 P (X
=p
= p
x
u=0 (1
x
u=0 (1
= u) =
p)u p
x+1
p)u = p 1(1(1p)p)
1
1(1p)x+1
p
=

APPM 4/5520
Problem Set One (Due Wednesday, September 4th)
Note: We spent week 1 in prerequisite review mode. So, for most of you this assignment will be pretty trivial. This course will ramp up quickly after week one, so things
will get interesting soon!

APPM/MATH 4/5520
Final Exam Review Problems
The nal exam is on Wednesday, December 18th in our normal classroom from 4:30 to 7pm.
It is not cumulative.
The exam will have 6 problems and you must choose and complete 5 out of 6 problems. There
is no grad

APPM/MATH 4/5520
Exam II Review Problems
Exam I: will be on Thursday, November 14th from 6:30 to 9:00pm in FLMG 155.
Optional Extra Review Session: will be on Wednesday, November 13th from 6 to 8 pm. This
location is still pending.
1. Let X1 , X2 , . . .

APPM/MATH 5520 Mathematical Statistics
Fall 2013 Exam Two, Take Home Part
Due Friday, December 6th
Welcome to the take-home part of exam II. This is an exam, so please do not discuss it
with anyone. Except me you are more than welcome to come talk to me a

APPM/MATH 4/5520
Exam I Review Problems
Exam I: will be on Thursday, October 4th from 6:30 to 9pm in a room TBA.
Optional Extra Review Session: will be on Wednesday, October 3rd from 6 to 8 pm in a room
TBA.
The actual exam will have around 6 problems. Th

The Central Limit Theorem
If X1 , . . . , Xn is a random sample from a distribution with mean and variance 2 < , then the limiting distribution of
n(X )
Zn =
D
is standard normal. That is, Zn Z N (0, 1) as n .
Note:
the CLT requires a nite variance (whic

APPM/MATH 5520 Mathematical Statistics
Fall 2013 Exam One, Take Home Part
Due Friday, October 11th
Welcome to the take-home part of exam I. This is an exam, so please do not discuss it with
anyone. Except me you are more than welcome to come talk to me ab