University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Midterm 1
CRN: 30357
Instructor: Jenna G. Tichon
Date: Monday, May 25, 2015
Time: 10:45 am to 12:15 pm
Location: 205 Armes
Instructions:
- All answers must be properly
Functions of Two or More Variables Solutions
1. Let Z = |X Y |.
As all equally likely P (E) =
n(E)
n()
6/36
10/36
8/36
pz (z) = 6/36
4/36
2/36
0
for z = 0
for z = 1
for z = 2
for z = 3
for z = 4
for z = 5
otherwise
2. Let X=value on first ball and Y =valu
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Midterm 2
CRN: 30302
Instructor: Jenna G. Tichon
Date: Monday, June 15, 2014
Time: 10:45 am to 12:15 pm
Location: 205 Armes
Instructions:
- All answers must be properl
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Final Exam
Date: Friday, June 24, 2011.
Instructor: Jenna G. Tichon
Time: 9:00 am to 11:00 am
Instructions:
- Show as much detail as you can.
- Nonprogrammable and non
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Final Exam
CRN: 30302
Instructor: Jenna G. Tichon
Date: Thursday, June 25, 2015
Time: 1:30-4:30 p.m.
Location: 204 Armes
Instructions:
- All answers must be properly j
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Final Exam
CRN: 30357
Instructor: Jenna G. Tichon
Date: Thursday, June 26, 2014
Time: 9:00 a.m.
Location: 207 Buller
Instructions:
- All answers must be properly justi
Chapters 1 and 2: Basic Concepts
1
Page 1
Basic Concepts
1.1
Set Theory
Definition 1. A set is a collection of distinct elements. If B is a set and b is an element of B, we write
b B; if b is not in B, we write b
/ B. We use cfw_ to denote the empty set
Final Exam Review
Page 1
Example 1. A game called Poker dice is played by rolling 5 six-sided fair dice at once. Find the following
probabilities.
(a) P r(one pair)
(b) P r(two pair)
(c) P r(full house)=P r(three of a kind and a pair)
Solution:
(a) P r(on
University of Manitoba
CRN:
Course: STAT 2400, Probability I
Date:
Time:
Another Sample Final Exam
Instructor: K. Davies
Location:
Page: 1 of 11
Department of Statistics
STAT 2400
Probability I
Another Sample Final Exam
Instructions:
- All answers mus
STAT 2400 Quiz 1
May 11th, 2015
Name:
Student Number:
Instructions:
Complete all 4 questions.
Show all of your work.
No outside materials.
1. (4 marks) Prove that
(A B) (A C) A (B C)
.
2. At a blood donation clinic, 70% of people are repeat donors, 65%
STAT 2400 Quiz 3
June 3rd, 2015
Name:
Student Number:
Instructions:
Complete all 4 questions and show all of your work.
No outside materials. Calculators are permitted.
1. An upper-level statistics course has 10 graduate students and 14 undergraduate st
STAT 2400 Quiz 2
May 20th, 2015
Name:
Student Number:
Instructions:
Complete all 4 questions and show all of your work.
No outside materials. Calculators are permitted.
1. An English class has 15 students: 7 males and 8 females. How many ways
can we lin
Joint PMF Solutions
1. X= number customers in half an hour
Y = number of customers that buy a cupcake
X P ( = 6)
Y ?
Y |X Bin(n = X, p = .3)
a) px,y (x, y) = px (x)py|x (y)
6 x
e 6 x y
.3 (.7)xy
px,y (x, y) =
x! y
0
for x = 0, 1, 2, . y = 0, 1, 2, . y x
Functions of Two or More Variables Examples
1. Suppose you roll two dice. Let X be the value of the first die and Y be the value of the
second die. Find the pmf of the absolute difference between X and Y (i.e. |X Y |).
2. A bucket has balls labelled 2, 2,
Probability Questions
1. Amy has 7 textbooks, 5 cookbooks, and 12 novels on her bookshelf.
(a) Whats the probability the first book is a textbook?
(b) Whats the probability the first book is a textbook or a cookbook?
2. Each day Jimmy goes to 7-11 to buy
1. Customers enter a bakery following a Poisson distribution with an average of 6 per half hour. Let X be the number of customers that enter
in half an hour. Let Y be the number of customers in half an hour
that buy a cupcake assuming people buy cupcakes
Functions of Two or More Variables Examples
1. Suppose you roll two dice. Let X be the value of the first die and Y be the
value of the second die. Find the pmf of the absolute difference between
X and Y (i.e. |X Y|).
2. A bucket has balls labelled 2, 2,
Line and Circle Permutation Problems
Amy is lining up a group of 8 friends so that she can take a picture of
them (she wont be in the picture). How many ways can this be done if:
a) Sammy and Jimmy need to have at least two people in between them?
b) Eri
STAT 2400 Quiz 4
June 10th, 2015
Name:
Student Number:
Instructions:
Complete all 4 questions and show all of your work.
No outside materials. Calculators are permitted.
1. Angela is really bored and is trying to make her little sister guess the suit of
Department of Statistics
STAT 2400
Probability I
Sample Final Exam
Instructions:
- All answers must be properly justified. Show as much detail as you can.
- Nonprogrammable and nongraphical calculators are allowed.
- This is a closed book exam: no books,
University of Manitoba
CRN:
Course: STAT 2400, Probability I
Date:
Time:
Another Sample Final Exam
Instructor: K. Davies
Location:
Page: 1 of 11
Department of Statistics
STAT 2400
Probability I
Another Sample Final Exam
Instructions:
- All answers mus
Final Exam Review
Page 1
Example 1. A game called Poker dice is played by rolling 5 six-sided fair dice at once. Find the following
probabilities.
(a) P r(one pair)
(b) P r(two pair)
(c) P r(full house)=P r(three of a kind and a pair)
Example 2. A survey
Violation of Assumptions
It should be noted that, in the previous two examples, the
departures from our assumptions would likely have been
noticed before the regression was even run.
There are remedies in situations such as these to correct for
nonlineari
Residuals
residual = actual value of y predicted value of y
A positive residual indicates that an observation falls
above the regression line and a negative residual indicates
that it falls below the line. As an example, check that
the residual for Studen
Some variability in Y is accounted for by the fact that,
as X changes, it pulls Y along with it. The remaining
variation is accounted for by other factors (which we
usually dont know).
The value of r2 has a special meaning in least-squares
regression. It
Examining Relationships
In statistics, we often encounter problems which
involve more than one variable. We often want to
compare two (or more) different populations with
respect to the same variable.
We use two-sample or ANOVA inference methods to
compar
Correlation
The correlation r measures the direction and strength
of a linear relationship between two quantitative
variables.
Suppose the values of two quantitative variables X
and Y have been measured for n individuals. Then
Association vs. Causation
We
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Supplementary Problems: Module 1
(Available on September 9th )
A. Problems from the textbook
From Chapter 1: 18, 19, 50, 52, 53, 55, 62, 63
From Chapter 2: 4, 8, 11, 1
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Supplementary Problems: Module 4
(Available on October 15th )
A. Problems from the textbook
From Chapter 5:
7, 11, 28, 42, 45, 49, 50, 61, 65, 66, 68 (a, b, c), 78, 81
University of Manitoba
Department of Statistics
Statistics 2400
Introduction to Probability I
Answers to Supplementary Problems from Module 4
Supplementary Problems from the textbook
From Chapter 5:
5.42.
a) 0.1613
b) 0.3325
d) 0.8213
e) X B(5, 0.67)
5.66
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Supplementary Problems: Module 3
(Available on October 7th )
A. Problems from the textbook
From Chapter 4: 2, 3, 12, 19, 22, 24, 28, 31, 33, 34, 49, 51, 56, 66, 73, 79
University of Manitoba
Department of Statistics
STAT 2400
Introduction to Probability I
Supplementary Problems: Module 2
(Available on September 9th )
A. Problems from the textbook
From Chapter 3: 3, 4, 11, 15, 17, 19, 21, 30, 36, 41, 42, 62, 76, 91, 93,