Probability and Statistics
Homework 2
Due: Wednesday 9/19/2012.
1. An academic department with ve faculty members narrowed its choice for department head
of either candidate A or candidate B . Each member then vote on a slip of paper for one of
the candid
Raskutti Statistics/Mathematics 309
Solution to Midterm 1
October 11, 2013
1. There are three fair 10-sided white dice with numbers 0, 1, 2, . . . , 9 and two fair regular 6-sided red dice with
numbers 1, 2, 3, 4, 5, 6. All five dice are rolled. The die r
Raskutti Statistics/Mathematics 309
Practice Final Solutions
December 11, 2013
Question 1
A 6-sided die is rolled repeatedly. The die rolls are mutually independent. Each die is uniformly likely to result
in a digit from 1,2,3,4,5, and 6.
1. (4 points) Wh
STAT/MATH 309
Solution 12
1
1
.
3
2
a) ( 35 ) 0.91
b) ( 37 ) 0.87
3
a) 1 ( 513 ) 0.08
b) 1 (1 (1)2 0.97
c) (1 (1)2 (1 (1)2 0.68
d) ( 12 ) 0.5 0.26
4
Assume we are considering a Geometric distribution with counting beginning at 0. M (t) =
1
where t < loge
Statistics/Mathematics 309
Raskutti
February 4, 2014
Assignment #3 Due Wednesday, February 12, 2014, by 5:00 P.M.
1. Follow the example proof for the inclusion-exclusion theorem
P (A B) = P (A) + P (B) P (A B)
and prove a three set version:
P (A B C) = P
Statistics/Mathematics 309
Raskutti
January 20, 2014
Assignment #1 Due Wednesday, January 29, 2014, by 5:00 P.M.
Turn in homework in lecture, discussion, or your TAs mailbox (just inside the main entrance of MSC, 1300
University Avenue)
1. Evaluate the fo
Statistics/Mathematics 309
Raskutti
January 28, 2014
Assignment #2 Due Wednesday, February 5, 2014, by 5:00 P.M.
Turn in homework in lecture, discussion, or your TAs mailbox. Please circle the discussion section you expect to
attend to pick up this assign
Statistics/Mathematics 309
Raskutti
February 18, 2014
Assignment #5 Due Wednesday, February 26, 2014, by 5:00 P.M.
Turn in homework in lecture, discussion, or your TAs mailbox (just inside the main entrance of MSC, 1300
University Avenue)
1. Events A and
Statistics/Mathematics 309
Raskutti
February 10, 2014
Assignment #4 Due Wednesday, February 19, 2014, by 5:00 P.M.
Turn in homework in lecture, discussion, or your TAs mailbox (just inside the main entrance of MSC, 1300
University Avenue)
1. An urn contai
Introduction to Probability Lecture Notes
Version April 28, 2015
David F. Anderson
Timo Seppalainen
Benedek Valko
c Copyright 2015 David F. Anderson, Timo Seppalainen and Benedek Valko
2000 Mathematics Subject Classification. Primary
Contents
Preface
1
Ch
Spring 2017
Stat 309
Homework 2 (100 points total) Solution
1. Events A, B and C are such that
P (A) = 0.7, P (B) = 0.6, P (C) = 0.5, P (AB) = 0.4, P (AC) = 0.3, P (BC) = 0.2, P (ABC) = 0.1.
Find:
(a) either B or C happens.
(b) at least one of A, B, C hap
Raskutti Statistics/Mathematics 309
Solution to Practice Midterm 2
April 1, 2014
1. Random variables X and Y have joint density fX,Y (x, y) = cxy for x > 0, y < 2, 2x < y and 0 otherwise.
(a) (5 points) Find c.
RR
Solution: Need to check that
f (x, y)dxdy
Statistics 309, Exam I
First Name:
Last Name.:
Be sure to show all relevant formulas and work !
1. (30) A survey on movies A, B, and C was conducted, and the table below lists the survey results on
the percentages of people in a city who watched movies A,
Statistics 309, Exam I
First Name:
Last Name.:
Be sure to show all relevant formulas and work !
1. (10 points) In a college 30% of students have cars. Students are selected at random to check if they
have cars.
(a) Find the probability that nine students
Homework 2 Solution
1.
2.
3.
4.
(a) For each position, there are 10 digits to choose from. Since each password contains 4
digits, the total number of different password are: 10^4
(b) There are 10 digits, so there are 10 cases where all four digits are the
Probability and Statistics
Homework 3
Due: Wednesday 9/26/2012.
1. One box contains six red balls and four green balls, and a second box contains seven red balls
and three green balls. A ball is randomly chosen from the rst box and placed in the second
bo
Homework 3 Solution
1.
(a) This problem can be understood in two ways:
(1) P(red from 1st and red from 2nd ):
P(red from 1st and red from 2nd ) = P(red from 2nd | red from 1st )*P(red from 1st )
= 6/10 * 8/11 = 0.436
(2) P(red from 1st) and P(red from 2nd
Probability and Statistics
Homework 4
Due: Wednesday 10/3/2012.
1. Airlines sometimes overbook ights. Suppose that for a plane with 50 seats, 55 passengers
have tickets,. Dene the random variable Y as the number of ticketed passengers who actually
show up
Homework 4 Solution
1.
a. In order for the flight to accommodate all the ticketed passengers who show up, no more
than 50 can show up. We need y 50.
P(y 50) = .05 + .10 + .12 + .14 + .25 + .17 = .83
b. Using the information in a. above,
P(y > 50) = 1 P(y
Probability and Statistics
Homework 5
Due: Wednesday 10/10/2012.
1. When circuit boards used in the manufacture of compact disc players are tested, the long-run
percentage of defectives is 5%. Let X = the number of defective boards in a random sample
of s
STAT/MATH 309
Solution 7
1
Let p =
Qn
(a) 1,
(b)
i=1
1
2
(1 pi ).
and
1
2
1
(x+1)(x+2)
(c)
2
(a) 1
(b) 0
(c) 1/2
(d) 1/2
(e) 0.4
3
(a) e16
(b) 1 e80
(c) e8 e40
(d) 1 e16
(e) e8a
4
P (X t + m X t)
P (X t)
P (X t + m)
=
P (X t)
P (X t + m|X t) =
1
For a Ge
STAT/MATH 309
Solution 8
1
P
(a) F (x) = 1 exp( ni=1 i x) for x > 0.
P
P
(b) f (x) = ni=1 i exp( ni=1 i x) for x > 0.
Q
(c) F (x) = ni=1 (1 ei x ) for x > 0.
P
Q
(d) f (x) = ni=1 i exp(i x) j6=i (1 ej x ) for x > 0.
2
(a) F (x) = 1 (1 x)n for 0 < x < 1, f