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
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 >
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
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
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) = ()()
Stat 426 Exam 1 Solution
Winter 2016
Problem 1
Of all customers purchasing Brand X fire alarms, 65% purchase a combined model that is also a
carbon monoxide detector. To find the probability that the
Homework 4
Moulinath Banerjee
University of Michigan
November 7, 2016
Problem 1: Consider the following model: Yi = 0 + 0 Xi + i for i = 1, 2, . . . , n where for
the moment the Xi s are fixed numbers
Stat 426 Winter 2016
Homework 4 Solution
Problem 1
Suppose a randomly chosen Stat 426 students scores (appropriately scaled) on Exam 1 (X)
and Exam 2 (Y) have a joint pdf
, (, ) =
a) Find k.
Thus
1
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
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
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 yo
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
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)
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) =
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 sampl
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 )
Homework 5 Solution
Stat 426 Winter 2016
Review Lectures 4 & 5, Secs 2.3, 3.7, 4.5. Practice problems: Q 7, 31, 81, Chap 4
Problem 1. Find the density of = ( ), = 1, , where ~( ) and , = 1, , are
inde
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 rando
Stat 426 Exam 2 Solution
Winter 2015
Problem 1 [Show working, steps, explanations] 10 points
Suppose ~(, 2 ), > 0.
(a) Use the pdf of X to find (not state) ()
!
=
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 s
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 t
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 ra
Homework 4
Practice problems: Chapter 6, Questions 3, 4; Chapter 7, Questions 3, 5, 11
Problem 1 (Chapter 6, Question 5) Show that if ~ , then 1 ~ ,
Note: Use the transformation method.
If ~ , then
Le
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 para
11.9
Example 11.8.2
369
Factorial Experiments
Sample size to achieve specied power
Suppose a test of the null hypothesis H0 : cfw_i all equal is required to detect a difference in
the treatment effect
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 fact
Homework 3. Due Feb. 5, 2:40pm In Class
Work out all these problems.
You must write down intermediate steps in order to reach the final answer.
1. [10pts] Rice 3.16
2. [10pts] Rice 3.18
3. [10pts] Let