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 w
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 I
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 + 2X
MIDTERM 1 SOLUTION
1.
z2
1
fZ (x) = e 2
2
(yz)2
1
fY |Z (y|x) =
e 22
2
2
1 z22 (yz)
2 2
fY,Z (y, z) = fY |Z (y|z)fZ (z) =
e
2
Let T = Y + Z.
Z
2
1 z22 (t2z)
2 2
fT (t) =
e
dz
2
Z
(z 22t )2
t2
1
ex
MIDTERM 2 SOLUTION
1. EWn = EXi2 = 2 , EWn2 = EXi4 = 3 2 < .
Xi are iid, so Wn 2 almost surely. (SLLN)
Since g(x) = log x is continuous, log Wn log 2 almost surely.
So, 12 log Wn log almost surely.
2.
STAT510 Mathematical Statistics I, Fall 2017
Homework 3: Due at 3pm, Friday, November 3, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. CB7.38
Solution: (a)
f (
x|) =
n
n
Y
xi
1
STAT510 Mathematical Statistics I, Fall 2017
Homework 6: Due at 3pm, Friday, November 17, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. Derive the explicit expression for the Pi
STAT510 Mathematical Statistics I, Fall 2017
Homework 6: Due at 3pm, Friday, November 17, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. Solution: (i)
R
Q
ni=1 f (xi )d
() = R Q
STAT510 Mathematical Statistics I, Fall 2017
Homework 7: Due at 3pm, Monday, December 11, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. CB5.42
2. CB5.38(a,b)
3. Consider a seque
STAT510 Mathematical Statistics I, Fall 2017
Homework 5: Due at 3pm, Friday, November 3, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. CB7.38
2. Suppose that X1 , , Xn are iid w
FINAL EXAM SOLUTION
1. X Ponsson(), Y Ponsson(), and X and Y are independent.
So, X + Y Ponsson( + ).
P (X = x, X + Y = w) = P (X = x, Y = w x) =
P (X = x|X + Y = w)
x e (wx) e
x! (w x)!
P (X = x, X +
STAT 510 Homework 6
1. C&B book 6.7, 6.10, 6.13, 6.14, 6.15, 6.16, 6.17, 6.19 6.20
2. Let Xi be iid from a location-scale family. Define M = sample median and Q1
and Q3 to be the first and third quart
STAT 510
Fall 2016
Midterm Solutions
1, Pls refer to Homework 1 Problem 2.
2, Pls refer to Homework 2 Problem 3.
3, Pls refer to Homework 3 Problem 4.
4, (1) N Poisson(100).
Let Xi be the amount that
STAT 510 Midterm 2
Friday, Nov 6, 2015
10am -10:50am
Last Name:
First Name:
Instructions: Please write your solution in the space provided. You may use any results
from class or homework as long as yo
STAT 510 Final
Monday, Dec 14, 2015
8am -11am
Last Name:
First Name:
Instructions: Please write your solution in the space provided. You may use any results
from class or homework as long as you state
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(, )
STAT 510 Midterm 1
Friday, Oct 2, 2015
10am -10:50am
Last Name:
First Name:
Instructions: Please write your solution in the space provided. You may use any results
form class or homework as long as yo
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 o
STAT510 Mathematical Statistics I, Fall 2017
Homework 8: Due at 3pm, Friday, December 15, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. For the random sample X1 , . . . , Xn fro
STAT510 Mathematical Statistics I, Fall 2017
Homework 3: Due at 3pm, Wednesday, October 4, 2017
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose that X1 , X2 are iid N (, 1), wh
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 7 Solutions
Fall 2014
Problem 1: (CB 5.42)
Solution:
(a)
x
)
n
x
= 1 [P (X1 < 1 )]n
n
x n
= 1 (1 )
n
P (
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 combinatio
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 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 I
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 asymptot
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 (
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 Il
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 th
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 8 Solutions
Fall 2014
Problem 1: For the random sample X1 , , Xn from N (, 2 ), find the asymptotic dist