Saturday, February 4, 2012
Page 1
Mathematics 1228B
Test 1
CODE 111
PART A (17 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
1
mark
1. In a group of 85 students, it is fo
Mathematics 1228A Assignment 1
(Due by 12 noon EDT Friday May 22, 2015)
Reminder: You must show sucient work to justify your answers.
1. Consider the set containing all rst year students at UWO. Let S be the set of all students
taking one or more courses
1
Math 1228A - Summer 2015
Assignment 1 Solutions
Question 1
We are thinking in this question only about rst year students at UWO. We have sets S and C which contain
all the students who are taking any Science courses, or any Social Science courses, respe
Extra Practice Questions for Section 1.1
1. Let T be the set of all people who subscribe to TV Guide, M be the set of all people who
subscribe to Macleans magazine, and N be the set of all people who subscribe to Newsweek.
(a) Write an expression in terms
Math 1228A/B Online
Lecture 1:
Using Sets
(text reference: Section 1.1, pages 1 - 2)
c V. Olds 2008
1
Techniques of Counting
Counting seems pretty basic. What were going to be learning about is how to count the number
of ways in which something can be don
166
4
CONTINUOUS RANDOM VARIABLES
Theorem: Let X be a discrete r.v. whose possible values are evenly spaced k units
apart, and let continuous r.v. Y be a good approximation for X. Let FX (x) and FY (y)
denote the cumulative distribution functions of X and
4.3 Normal Random Variables
179
Example 4.11. (e) Find a probability expression in terms of X which corresponds to P r[1 < Z < 2].
Since Z =
X25
5 ,
then if Z > 1, we have
X 25
> 1 X 25 > 5 X > 30
5
(That is, since = Z + , then Z > 1 X > 5(1) + 25 = 30.)
186
4
CONTINUOUS RANDOM VARIABLES
Approximating the Binomial Distribution
Recall: A random variable X which counts the number of successes in a series of n
Bernoulli trials in which the probability of success is p is a Binomial r.v., B(n, p), with
mean =
176
4
CONTINUOUS RANDOM VARIABLES
Example 4.8. (c) P r[0.50 < Z < 0.50]
Once again, we will need to use symmetry and also complementation.
P r[0.50 < Z < 0.50] = P r[Z < 0.50] P r[Z < 0.50]
= P r[Z < 0.50] P r[Z > 0.50]
= P r[Z < 0.50] (1 P r[Z < 0.50])
=
182
4.4
4
CONTINUOUS RANDOM VARIABLES
Approximately Normal Discrete Random Variables
(Note: Thats not the title the text uses for this section.)
Many discrete r.v.s are approximately normal. What does that mean?
Denition: If discrete r.v. X, with mean and
100
2.6
2 PROBABILITY
Bayes Theorem
Consider a stochastic process in which there are 2 experiments. In the rst experiment, outcomes A,
B and C occur with probabilities 1/2, 1/5 and 3/10 respectively. In the second experiment, event
D either does or does n
Saturday, February 4, 2012
Page 1
Mathematics 1228B
Test 1
CODE 111
PART A (17 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
1
mark
1. In a group of 85 students, it is fo
Saturday, February 2, 2013
Page 1
Mathematics 1228B
Test 1
CODE 111
PART A (18 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
1
mark
1. Let U = cfw_1, 2, 3, 4, 5, 6, A = c
4.1 Probability Density Functions
163
The width of the rectangle and the base of the triangle are both given by 3 1 = 2. The height
1
of the rectangle is f (1) = 1 . The height of the triangle is the vertical distance from 8 to f (3), so
8
3
1
2
1
1
the h
Math 1228A/B Online
Lecture 29:
The Standard Normal Random Variable Z
(text reference: Section 4.2)
c V. Olds 2008
172
4.2
4
CONTINUOUS RANDOM VARIABLES
The Standard Normal Random Variable
There is a very important family of continuous r.v.s which are cal
140
3 DISCRETE RANDOM VARIABLES
Therefore we need to nd E(X) and (X). (We could nd the pdf of Y and nd E(Y ) and (Y )
directly, but the calculations for E(X) and (X) are easier, i.e. involve easier numbers.)
We are told that the probability that George se
3.3 The Mean and Standard Deviation
131
We see that the probability distribution of X is:
x
1
2
3
4
We get:
=
1
3
+ 2
2
10
12
9
4
+
+
+
20 20 20
7
=
4
1
E(X) =
=
P r[X = x]
1/2
3/10
3/20
1/20
10
20
35
20
+ 3
3
20
+ 4
1
20
Therefore, the expected number of
3.3 The Mean and Standard Deviation
137
in which the probability of success is p = 1/6 each time. Therefore we have X = B(300, 1/6). Of
course, the mean of X = B(300, 1/6) is given by
= np = 300
1
6
= 50
Also, the variance of X = B(300, 1/6) is given by
2.7 Independent Trials
113
Let E1 , E2 and E3 be the events that a type 1 urn, a type 2 urn or a type 3 urn is chosen,
respectively. We dont know how many urns there are, but we do know the percentage of the total
which are of each of the various types. O
3.1 Probability Distributions and Random Variables
121
There are certain properties which a cumulative distribution function, F (x), must always have.
You will have noticed some of these already, while we were nding F (x) in the examples. In order
to expr
Math 1228A/B Online
Lecture 25:
Independence of rvs and
The Joint Distribution of X and Y
(text reference: Section 3.4, pages 136 - 139)
c V. Olds 2008
146
3.4
3 DISCRETE RANDOM VARIABLES
Independent Random Variables
Example 3.23. Recall Example 3.17. Fin
150
3 DISCRETE RANDOM VARIABLES
Notice: In the joint distribution table, the sum of all the entries in (the main body of) the table is
1. And the probabilities we already lled in sum to 1, so all the other table entries must be 0.
Also Notice: We already
3.4 Independent Random Variables
153
sample point
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
x
1
1
1
1
0
0
0
0
y
2
1
1
0
2
1
1
0
w = xy
2
1
1
0
0
0
0
0
5
1
Of course, since each sample point occurs with probability 8 , we get P r[W = 0] = 8 , P r[W = 1] =
1
and P r[
157
4
Continuous Random Variables
4.1
Probability Density Functions
We dened a discrete random variable to be one which has a nite number of possible values. Some
r.v.s can take on any real value, perhaps within a limited range. This is often true of a ra
3.1 Probability Distributions and Random Variables
127
Next, consider the following:
Fact: Selecting a small random sample from a very large population can be considered
as performing independent trials.
Notice: If the population is not very large then th
1
Math 1228B Final Exam, April 22 2013
Answers Code 111
Part A
1.
A
6.
A
11. E
16. B
21. B
26. E
31. B
2.
7.
12.
17.
22.
27.
32.
E
C
B
B
A
C
C
3.
8.
13.
18.
23.
28.
33.
D
E
C
D
E
B
A
4.
9.
14.
19.
24.
29.
34.
A
D
A
B
A
D
E
5.
10.
15.
20.
25.
30.
35.
B
C
C
1
Math 1228B Final Exam, April 16 2014
Answers Code 111
Part A
1.
B
6.
D
11. C
16. E
21. B
26. D
31. D
2.
7.
12.
17.
22.
27.
32.
C
A
B
E
A
C
D
3.
8.
13.
18.
23.
28.
33.
D
C
B
D
A
E
E
4.
9.
14.
19.
24.
29.
34.
C
B
A
A
B
E
C
5.
10.
15.
20.
25.
30.
35.
E
E
D
April 16, 2014
Page 1
Mathematics 1228B
Final Examination
CODE 111
PART A (35 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
A1. On a January morning, it was noticed that
Saturday, March 16, 2013
Page 1
Mathematics 1228B
Test 2
CODE 111
PART A (18 marks)
1. [ 1 mark ] A box contains 3 dimes (10 cent coins) and 3 quarters (25 cent coins). Four coins are selected
at random from the box, without replacement. Which one of the
Saturday, February 4, 2012
Page 1
Mathematics 1228B
Test 1
CODE 111
PART A (17 marks)
1. [ 1 mark ] In a group of 85 students, it is found that 60 students have a pen, 50 students have a pencil,
and 25 students have a pen but not a pencil. How many studen
Saturday, March 17, 2012
Page 1
Mathematics 1228B
Test 2
CODE 111
PART A (18 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
1
mark
1. If P r[A] = .5, P r[A B] = .3 and P r
Students Name [print]
Student Number
Mathematics 031 Test 3 CODE 111
45 minutes
1
mark
Wednesday, February 1, 2006
1. Find the number of different ways to give 30 cents in change using quarters, dimes or
nickels if coins of a larger value are given before
Saturday, March 12, 2011
Page 1
Mathematics 1228B
Test 2
CODE 111
PART A (18 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
1
mark
1. If P r[A B] = .2 and P r[A B c ] = .5