Homework 4 (Due 6-10)
1) Let Y1 , . . . , Yn be iid N (, 2 ). Let N (, 2 ) and treat 2 , , and 2 as fixed and
known. Find the posterior distribution p(|y). This distribution will depend on 2 , ,
and 2 . Use the hints given in class.
2) Let Y1 , . . . , Yn
1
HW1 Solution
Prob 1
)i+j /(1
a) P (W > i + j |W > i) = P (W > i + j )/P (W > i) = (1
P (W > j )
)i = (1
)j =
b) Considering the memoryless property, we have for any integer n > 1
P (W > n) = P (W > 1) P (W > n
1).
Then inductively, we have P (W > n) =
Lecture 5 Stat 426
Moments and Expectations
Sec 4.1-4.3, 4.5
1. Review
() = () ()
1.1 If X is a discrete RV with pmf f and range R , and Y=g(X), then
provided the sum converges absolutely.
E(Y) = ()()
1.2 If X has density f, and Y=g(X), then
provided that
Stat 426 Winter 2016, Homework 2
Selected practice problems (no solutions provided, not for submission or grading):
Chap 2, Q4, 7, 8, 11, 31; Chap 4, Q 2, 3; Review Lecture 2
Turn in the solutions to the following questions:
Problem 1. Chap 2. Q 14
Two bo
Stat 426 W2014
Homework 1 Solution
Problem 1
A lie detector test has a 90% detection rate if a person is lying, and a 5% detection rate if a
person is not lying. Suppose that 1% of all individuals in a certain population tell lies. Two lie
detector tests
2009 Fall Stat 426 : Homework 1
Moulinath Banerjee University of Michigan Announcement: The homework carries 50 points and contributes 5 points to the total grade. Your score on the homework is scaled down to out of 5 and recorded. 1. A geometric random v
Some Inequalities and the Weak Law of Large Numbers
Moulinath Banerjee
University of Michigan
August 30, 2012
We rst introduce some very useful probability inequalities.
Markovs inequality: Let X be a non-negative random variable and let g be a
increasing
Fall 2014, Stat 426 : Homework 1
Moulinath Banerjee
University of Michigan
Announcement: The homework carries 80 points and is due on Sept
29th in class. It contains two supplementary problems that you dont
need to turn in.
1. A geometric random variable
1
HW2 Solution
Prob 1
(i) If we denote the number of heads based on the later N tosses as N1 , the total number of
heads will be N + N1 . And,
1
1
3
E (N + N1 ) = E (N ) + E (N1 ) = n + E (E (N1 |N ) = n + E (0.5N ) = n
2
2
4
And by Theorem B on Pg151 of
Stat 426 : Homework 3.
Moulinath Banerjee
October 28, 2012
Announcement: The homework carries a total of 41 points. The maximum possible score is
40 points.
1. (a) Consider the standard estimator of 2 based on X1 , X2 , . . . , Xn i.i.d. N (, 2 ). This i
Stat 426 Winter 2016
Homework 7 Solution
Problem 1 Question 1 Chapter 7
Consider a population consisting of five values, which are 1,2,2,4,8. Find the population mean
and variance. Calculate the sampling distribution of the mean of a sample of size 2 by g
Midterm 1, Winter 06
Moulinath Banerjee
University of Michigan February 22, 2006
Announcement: The exam carries 30 points but the maximum you can score is 25. (1) If X and Y are two uncorrelated random variables, are they necessarily independent? On the o
Stat 426
Solutions for Homework 3
1. Let Zi = Xi Yi . Thus, E[Z] = 1
central limit theorem,
(X
2 and V ar(Z) =
Y ) (1 2 )
Z (1
p
=p 2
2
2
( 1 + 2 )/n
( 1+
2 )
2
2 )/n
2
2
1+ 2
n
. By the
d
! N (0, 1)
2. Since X1 , . . . , X5 0 N (6, 0.2) i.i.d., 4(Xi )2 =
1
HW1 Solution
Prob 1
a) P (W > i + j|W > i) = P (W > i + j, W > i)/P (W > j) = P (W > i + j)/P (W > i) =
(1 )i+j /(1 )i = (1 )j = P (W > j).In this derivation, we have used the fact that
P (W > k) = (1)k for any integer k 0, from the properties of the ge
Stat 426 Lecture 1:
Review of Probability
1
Chapter 1
Introduction
A sound background in probability is essential to understanding the theory
of statistics
2
The Essentials
is the sample space, the set of all possible outcomes of a random experiment. The
Stat 426 Winter 2016
Homework 8 Solution
Chapter 8, Q 5, 19, 23, 31, 37 Review lectures 9 and 10
Practice problems: Chapter 8, Questions, 1, 3, 19, 23, 31 (Not for submission)
Problem 1. The Beta distribution is useful for modeling random variables that a
Homework 1 (Due 5-13)
Instructions: Turn in at the beginning of class. Late homework not accepted.
1) Evaluate
x4 e2x dx
0
No need to show any work. Relate to the gamma distribution.
2) Suppose the random variable X has the density given by
fX (x) =
a) Fi
Stat 426 Fall 2014
Homework 2
Let () =
, 1 1 , and () = 0 otherwise, where 1 1. Show that
Problem 1. Chap 2, Q34
1+
2
() is a density and find the corresponding cdf. Find the quartiles and median of the
distribution in terms of .
() 0 and () =
1
1
F(x)=
Homework 6
Practice problems: Chapter 8, Q 5, 19, 23, 31, 37
1. The Beta distribution is useful for modeling random variables that are restricted to the interval [0,1].
The Beta distribution with parameters (a,b), and its expected value and variance are g
Fall 2015, Stat 426 : Homework 1
Moulinath Banerjee
University of Michigan
Announcement: Homework to be turned in 6th October to GSI. Time
and place will be announced.
1. Problems from Book. Chapter 2: 24, 32, 37, 44, 71. (50 points) Also
starred problems
Stat 426 : Homework 2 solutions.
Moulinath Banerjee
University of Michigan
October 18, 2005
Announcement: For purposes of the homework, you can cite any results in the handouts or
the text-book or any others proved in class, without proof. The homework ca
379
Exercises
(a) Initial block: 1, 3, 4; v
7.
(b) Initial block: 1, 2, 4, 8; v
(c) Initial block: 1, 2, 4; v
8.
5.
5. Balanced incomplete block design
Consider an experiment to compare 7 treatments in blocks of size 5. Taking all possible
combinations of
11.9
371
Factorial Experiments
step frequency is increased, but that the linear trend is not the same for the two step heights.
The experimenters wanted to examine the average behavior of the two factors, so despite
this interaction, they decided to exami