Homework 7 - Solution
(p.325: 6.59)
Show that if Y1 has a 2 distribution with v1 degrees of freedom and Y2 has a 2 distribution
with v2 degrees of freedom, then U = Y1 + Y2 has a 2 distribution with v1 + v2 degrees of
freedom, provided that Y1 and Y2 are
Fall 2015
Stat 310
Homework 2
Due: 10/06 (Tuesday) 4:00pm before class
1. (10pts) A large sample is taken from an asymptotic distribution with a long left tail.
(a) Which one will be probably be larger mean or median?
(b) Will the sample skewness probably
Homework 6 - Solution
(p.364: 7.9)
Refer to Example 7.2. The amount of fill dispensed by a bottling machine is normally
distributed with =1 ounce. If n = 9 bottles are randomly selected from the output of the
machine, we found that the probability that th
Fall 2015
Stat 310
Homework 1 (100 points total)
Due: 09/17 (Thursday) 4:00pm before class
1. (12 pts) (Rice book Chapter 1 # 41) A drawer of socks contains seven black socks, eight blue
socks, and nine green socks. Two socks are randomly chosen in the da
Fall 2015
Stat 310
Homework 2
Due: 10/06 (Tuesday) 4:00pm before class
1. (10pts) A large sample is taken from an asymptotic distribution with a long left tail.
(a) Which one will be probably be larger mean or median?
(b) Will the sample skewness probably
Fall 2015
Stat 310
Homework 1 (100 points total)
Due: 09/17 (Thursday) 4:00pm before class
Solutions
1. (12 pts) (Rice book Chapter 1 # 41) A drawer of socks contains seven black socks, eight blue
socks, and nine green socks. Two socks are randomly chosen
Fall 2015
Stat 310
Homework 3
Due: 10/20 (Tuesday) 4:00pm before class
1. Textbook Chapter 7 Problem 8, Page 240.
Answer: For part a, using the CLT
P (| p| > ) = P (
p
n| p|
p
p(1 p)
>
n
p(1 p)
) = 0.025
p
Since n|p| is approximately a standard normal, on
Name
Campus ID
Stat 310: Introduction to Probability and Mathematical Statistics II, Spring 2015
Derek Bean
MIDTERM 2
Total: 50 points + 20 bonus points
Instructions: Read each problem carefully. There are some bonus questions throughout the test, so be
s
Stat 310: Introduction to Probability and Mathematical Statistics
II, Spring 2015
Derek Bean
Probability Review: Continuation of Lecture 3
Due: 1/26/2015
This is material from Rice Chapter 3 and 4 that I intended to review by
the end of Lecture 3 (held on
Name
Campus ID
Stat 310: Introduction to Probability and Mathematical Statistics II, Spring 2015
Derek Bean
PRACTICE MIDTERM 1
Total: points
FOR FULL CREDIT, SHOW ALL YOUR WORK.
1. Let cfw_x1 , x2 , . . . , xN be a P
finite population of N units. Suppose
Name
Campus ID
Stat 310: Introduction to Probability and Mathematical Statistics II, Spring 2015
Derek Bean
PRACTICE MIDTERM 1
Total: points
FOR FULL CREDIT, SHOW ALL YOUR WORK.
1. Let cfw_x1 , x2 , . . . , xN be a P
finite population of N units. Suppose
Name
Campus ID
Stat 310: Introduction to Probability and Mathematical Statistics II, Spring 2015
Derek Bean
PRACTICE MIDTERM 2
Total: points
FOR FULL CREDIT, SHOW ALL YOUR WORK.
1. Let X1 , . . . , Xn be a random sample. A 100(1 )% lower confidence bound
Name
Campus ID
Stat 310: Introduction to Probability and Mathematical Statistics II, Spring 2015
Derek Bean
MIDTERM 1
Total: 40 points
FOR FULL CREDIT, SHOW ALL YOUR WORK.
1. Let X1 , . . . , Xn be a simple random sample of a population, and let Y1 , . .
Assignment # 4 Stat 310 Due Friday, 9/30 by 4:00 P.M.
Turn in homework to your TAs mailbox using this sheet as the cover page.
Your TAs Name: Zifeng Zhao
Fill in your name and also circle the discussion section that you ATTEND. You will pick up the graded
Homework 11 - Solution
(p.409: 8.39)
Suppose that the random variable Y has a gamma distribution with parameters = 2 and an
unknown . In Exercise 6.46, you used the method of moment-generating functions to prove
a general result implying that 2Y/ has a 2
Assignment # 6 Stat 310 Due Friday, 10/14 by 4:00 P.M.
Turn in homework to your TAs mailbox using this sheet as the cover page.
Your TAs Name: Zifeng Zhao
Fill in your name and also circle the discussion section that you ATTEND. You will pick up the grade
Assignment # 3 Stat 310 Due Friday, 9/23 by 4:00 P.M.
Turn in homework to your TAs mailbox using this sheet as the cover page.
Your TAs Name: Zifeng Zhao
Fill in your name and also circle the discussion section that you ATTEND. You will pick up the graded
Homework 2 - Solution
(P.307: 6.2)
Let Y be a random variable with density function given by
(3 / 2) y 2 , 1 y 1,
f ( y)
0, elswhere.
a.
b.
c.
Find the density function of U1 = 3Y.
Find the density function of U2 = 3 - Y.
Find the density function of U3
Homework 3 - Solution
(p.338: 6.75)
Refer to Exercise 6.74. Suppose that the number of minutes that you need to wait for a bus is
uniformly distributed on the interval [0,15]. If you take the bus five times, what is the
probability that your longest wait
Homework 4 - Solution
(p.394: 8.1)
Using the identity
( ) [ E ( )] [ E ( ) ] [ E ( )] B( ) ,
Show that
MSE( ) E[( ) 2 ] V ( ) ( B( ) 2 .
Solution:
Let B B( ) . Then,
MSE( ) E[( ) 2 ]
E[( E ( ) B) 2 ]
E[( E ( ) 2 ] E ( B 2 ) 2 B E[ E ( )]
V ( ) B 2 .
(p
Homework 5 - Solution
(p.385: 8.13)
We have seen that if Y has a binomial distribution with parameters n and p, then Y/n is an
unbiased estimator of p. To estimate the variance of Y, we generally use n(Y/n)(1 Y/n).
(a) Show that the suggested estimator is
Homework 8 - Solution
(p.456: 9.15)
Refer to Exercise 9.3. Show that both 1 and 2 are consistent estimators for .
Solution:
First, E (1 ) E (Y ) 1 / 2 1 / 2 1 / 2 . By section 6.7, we can find the density function
n 1
of 2 : g n ( y ) n( y ) ,
y 1. Thus,
Homework 9 - Solution
(p.454: 9.17)
Suppose that X1, X2, ., Xn and Y1, Y2, ., Yn are independent random samples from populations with means
1 and 2 and variances 1 and 2 , respectively. Show that X Y is a consistent estimator of 1 - 2.
2
2
Solution:
By La
Homework 10 - Solution
(p.456: 9.19)
Let Y1, Y2, ., Yn denote a random sample from the probability function
y 1 , 0 y 1,
f ( y)
elsewhere,
0,
where > 0. Show that Y is a consistent estimator of /(+1).
Solution:
First, we know that Yi ~Beta(, 1). Thus E(
Homework 12 - Solution
(p. 462: 9.37)
Let X1, X2, ., Xn denote n iid Bernoulli random variables such that
P(Xi = 1) = p and P(Xi = 0) = 1 - p,
for each i = 1, 2, ., n. Show that i 1 X i is sufficient for p by using the factorization criterion
n
given in T
Name
Campus ID
Stat 310: Introduction to Probability and Mathematical Statistics II, Spring 2015
Derek Bean
PRACTICE MIDTERM 2
Total: points
FOR FULL CREDIT, SHOW ALL YOUR WORK.
1. Let X1 , . . . , Xn be a random sample. A 100(1 )% lower confidence bound