HW1
Solutions
Problem 1
2
1
Note that X1 X = 3 X1 3 X2 1 X3 and similarly for other deviations. Thus
3
2/3 1/3 1/3
A = 1/3 2/3 1/3 .
1/3 1/3 2/3
The covariance matrix is ACov(X)A = 2 AA = 2 A. Hence we get B = A.
Problem 2
(a) Note rst
MX (t) = E[etX ] =
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 1: Due on 3pm, Tuesday, Sep. 2, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose X is a 3 1 random vector with cov
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 3 Solutions
Fall 2014
Problem 1: Suppose that X1 , X2 are iid N (, 1), where is unknown. Let T = X1 + 2X2 be a
statistic. Show that T is not a sucient statisti
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 3: Due on 3pm, Wednesday, Sep. 24, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose that X1 , X2 are iid N (, 1),
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 2 Solutions
Fall 2014
Problem 1: [6 pts] Let (F ) to be the variance of the distribution F . (a) Write out the expression
of (F ); (2pts) (b) Find the expressi
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 2: Due on 3pm, Monday, Sep. 15, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Let (F ) to be the variance of the distri
HOMEWORK 3 SOLUTION
1. T = aX + bY.
If b 6= 0, Y = T aX
and X = X.
b
|det(
Z
(x, y)
1
)| = | |.
(x, t)
b
fT (t) =
fXY (x,
t ax 1
)| |dx.
b
b
If b = 0 and a 6= 0, T = aX. Then X = a1 T.
1
1
1
1 dx
fT (t) = fX ( t)| | = | |fX ( t) = | |
a dt
a
a
a
Z
1
fXY (
HOMEWORK 5 SOLUTION
1. Let g(x, y) = xy . Then
2
x
2y
g(x ,y )
x
g(x ,y )
1
y ,
y
= x2 .
y
2
+ y2 x4 2x,y x3 .
y
y
n
By Theorem 5.5.28, n( X
Yn
2 =
=
d
x
y )
N (0, 2 ).
d
2. n(Fn (x) F (x) N (0, F (x)(1 F (x). Let g(x) = x. Then g0(x) =
1 12
.
2x
By Del
STAT 510 Homework 5
1. See C&B book Theorem 5.5.28 for Multivariate Delta Method, then try the
following question: (X1 , Y1 ), . . . , (Xn , Yn ) are iid bivariate random vectors with
finite 2nd order moments. Let x = E(X1 ), y = E(Y1 ), x2 = var(X1 ), y2
HOMEWORK 3 SOLUTION
1. (a) Suppose X has distribution Exp(), then f (x) =
and x > 0,
P (X > t + x|X > t) =
e
t+x
t
1 x/
.
e
For any t > 0
x
= e = P (X > x).
e
So, the exponential distribution is memoryless. This relation implies that, conditioned that yo
STAT 510 Homework 1
1. Suppose Z has standard normal distribution.
(1) Use the mgf to find the first four moments of Z.
(2) Use the first part to deduce E(X 3 ) and E(X 4 ) where X Normal(, 2 ).
(3) Prove Z 2 2 (1).
2. Let Xm has pdf
xm1 ex/
m (m1)!
for
Fall 2015
STAT 510 Mathematical Statistics I
Homework 1 Solution
Problem 1
1) Z N (0, 1)
2
f (z) =
z
1 e 2
2
tz
+
Z
z2
1
etz e 2 dz
2
Z +
1
2
etzz /2 dz
=
2
Z +
1
1
2
2
=
e 2 (zt) +t dz
2
Z +
1
1
2
t2 /2
e 2 (zt) dz
=e
2
t2 /2
=e
Mz (t) = E(e ) =
E(Z m
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 5: Due on 3pm, Friday, Oct. 31, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. CB7.38
2. Suppose that X1 , , Xn are iid
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 4: Due on 3pm, Monday, Oct. 13, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose X1 , , Xn iid N (, 2 ), where bot
Some useful equalities!
Sum of geometric series: 1 + p+ p^2 + . + p^m = (1-p^cfw_m+1)/(1-p), when p<1. !
Gamma(n) = (n-1)!, !
Gamma(t+1) = t Gamma(t)!
!
The mean and variance of a linear combination of independent random variables!
HW2: 7; PP1: 5!
!
S
STAT510 Fall 2014
Mathematical Statistics I
Midterm Solutions
October 8, 2014, Wednesday, 10:00am11:50am
Student ID:
Name:
1. Please print your name and student ID number in the above space and circle the
discussion section number.
2. This is a closed-boo
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 8: Due on 3pm, Monday, Dec. 15, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. For the random sample X1 , , Xn from N (,
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 8 Solutions
Fall 2014
Problem 1: [12 pts] For the random sample X1 , , Xn from N (, 2 ), nd the asymptotic
distribution of mn = n1 n (Xj Xn )3 , where Xn is th
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 7 Solutions
Fall 2014
Problem 1: [10 pts] (CB 5.42)
Solution:
(a)
x
)
n
x
= 1 [P (X1 < 1 )]n
n
x n
= 1 (1 )
n
P (n (1 X(n) ) x) = P (X(n) 1
Let = 1/. Then we
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 7: Due on 3pm, Monday, Dec. 1, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. CB5.42
2. CB5.38(a,b)
3. Consider a sequen
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 6: Due on 3pm, Wednesday, Nov. 12, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. CB6.42
2. CB6.43
3. CB7.65
4. Assume t
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 6 Solutions
Fall 2014
Problem 1: [10 pts] (CB 6.42)
Solution:
x
(a) hg1 (x1 , x2 , ., xn ) = (cx1 + a, cx2 + a, ., cxn + a) W (hg1 (x1 , x2 , ., xn ) = c + a +
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 5 Solutions
Fall 2014
Problem 1: [6 pts] (CB 7.38)
Solution:
For both part (a) and (b), it can be shown that the distributions belong to the exponential family
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 4 Solutions
Fall 2014
Problem 1: [5 pts] Suppose X1 , , Xn iid N (, 2 ), where both and 2 are unknown
parameters. Let S 2 = (n 1)1 n (Xi X)2 be the sample vari
STAT 510 Homework 4
1. About the bivariate normal distribution
(a) Suppose Z1 and Z2 are independent standard normal random variables.
Find function X1 = h1 (Z1 , Z2 ) and X2 = h2 (Z1 , Z2 ) such that (X1 , X2 )
has bivariate normal distribution with para
STAT 510 Homework 3
1. Suppose (X, Y ) has joint pdf f (X, Y ), derive the pdf for T = aX + bY .
2. C&B Book, 4.4 (d); 4.16, 4.17, 4.19, 4.21, 4.24, 4.27
3. Let T and U be independent with T gamma(, ) and U gamma(, ). Let
X = T + U and Y = T /(T + U ). De
Chapter
3
Marginal Distributions and Independence
3.1
Marginal distributions
Given the distribution of a vector of random variables, it is possible in principle to
find the distribution of any individual component of the vector, or any subset of component
Chapter
5
Transformations: Jacobians
In Section 4.1, we saw that finding the pmfs of transformed variables when we start
with discrete variables is possible by summing the appropriate probabilities. In the
one-to-one case, it is even easier. That is, if g
STAT 510 Homework 2
1. Suppose we consider a Poisson process with occurrence rate , as in the notes.
(a) Show that the exponential distribution is memoryless (like the geometric).
Interpret this in terms of the Poisson process.
(b) Let T1 and T2 be the ti