Solutions to STAT 302 Homework No.1
Problem 1. (a) P(type O+ ) = 1/3.
(b) P(type O) = P(type O+ or type O ) = P(type O+ ) + P(type O ) = 1/3 + 1/15 = 6/15.
(c) P(type A) = P(type A+ or type A ) = P(type A+ ) + P(type A ) = 1/3 + 1/16 = 19/48.
(d) P(type A
Solutions to STAT 302 Homework No. 3
Problem 1. a). Let X be the number of monthly malfunctions, and C be the monthly costs of eld
representative visits, which is 40X . Then,
E (C ) = E (40X ) = 40E (X ) = 40(5) = $200.
V (C ) = V (40C ) = 402 V (X ).
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Stat 302 Assignment 1 (solution)
Q1.
(a) There are 7!=5040 possible dierent signals can be made, if there are no
restrictions.
(b) There are 2!2!3! arrangements such that the red ags are rst in line, then
the blue ags, then the green ags. Similarly, for e
CPSC 213
Introduction to Computer Systems
Winter Session 2014, Term 2
(0) Lecture 1 Jan 5
Introduction
The Basics
CPSC 213: Introduction to Computer Systems
What is a computer?
What is a computer system?
Why should you care?
How do you get an A?
a MAC
Mathematics 220 Homework 8  Solutions
1. Problem 1: Let f : A B, and let C1 ,C2 be subsets of A.
(a) Prove that f (C1 C2 ) = f (C1 ) f (C2 )
Solution:
Proof:
We prove the two inclusions (i) f (C1 C2 ) f (C1 ) f (C2 ) and (ii) f (C1 ) f (C2 )
f (C1 C2 ).
Mathematics 220 Homework 2  Solutions
1. (2.20) In each of the following, two open sentences P (x) and Q(x) over a domain S are given.
Determine all x S for which P (x) Q(x) is a true statement.
(a) P (x) : x 3 = 4; Q(x) : x 8; S = R
Solution: All x S fo
Mathematics 220
Solutions to Homework 4
3.2. Let n N. Prove that if n 1 + n + 1 1, then n2 1 4.
Solution:
Result. Let n N. If n 1 + n + 1 1, then n2 1 4.
Proof. Suppose that n N. Then n 1, so that n + 1 > 1. Since n 1 is
always nonnegative
Mathematics 220
Homework Set 10
1. 12.4
Solution: We need (n2 + 1)1  < , so
n2 + 1 >
n2 >
1
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1
1
1
1 .
1
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(n2 + 1)1 0 =
n2
1
1
< 2
+1
N +1
Hence the sequence converges to 0.
2. 12.6
Mathematics 220
Homework Set 9
Due: November 21
If you are using the 2nd edition, be careful question numbers may not agree.
10.20, 10.24
10.26, 10.28,
10.42 (draw a picture and think carefully about cases)
10.46 (induction is your friend)
EQ1 Let S,
MATH/STAT 302
Introduction to Probability
Midterm Exam 1
Section 201
Feb 15, 2007
Time: 50 minutes for working
Name:
Student No:
Instructions:
Read the whole exam paper carefully. We suggest you start with the
questions you think are easiest.
You must exp
Stat 302 Midterm 2
Thursday, March 17, 2016
Student Name _
Student Number _
You have exactly 50 minutes to complete this exam.
This test has 6 pages including this one.
Only nonprogrammable calculators are allowed for electronics.
That means no graphing
Stat 302 Midterm 1
Thursday, February 4, 2016
Student Name _
Student Number _
You have exactly 50 minutes to complete this exam.
This test has 6 pages including this one, and tables.
Only nonprogrammable calculators are allowed for electronics.
That mean
Winter 2016 STAT 302
Chapter 7 Properties of Expectation
Learning outcomes:
Aim: Apply properties of expectation to describe relationships between two
random variables, and to compute probabilities, expectation and variance
by conditioning.
Calculate the
Winter 2016 STAT 302
Chapter 6 Jointly Distributed Random Variables
Learning outcomes:
Aim: Describe the joint and conditional distributions related to two or
more discrete or continuous random variables
Recall the following definitions relating to two d
MATH 302
Sample midterm 1
1. (a) Dene carefully: A probability P on a given sample space S.
(b) events A, B, C are independent.
(c) Dene carefully: The conditional probability of an event A given
an event B of positive probability.
a. P is a function from
MID TERM EXAM
STAT 302
February 25, 2009
Time: 50 minutes
Student Name (please print): _ ._
SOLMTIONS
Student Number:
Notes:
0 This exam hasLl problems on $ pages1 including the cover. Please check to make sure
that you have all pages.
c There are a total
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http:/www.stat.ubc.ca/~bouchard/courses/stat302fa201516/
Intro to Probability
Instructor: Alexandre Bouchard
Fall 2015
Monday, October 5, 15
Plan for today:
Random variables, conditioning, and expectation
Some background for assignment 1
Monday, Octob
http:/www.stat.ubc.ca/~bouchard/courses/stat302fa201516/
Intro to Probability
Instructor: Alexandre Bouchard
Fall 2015
Wednesday, October 7, 15
Plan for today:
Wrapping up yesterdays problem
Graphical models
Wednesday, October 7, 15
http:/www.stat.ubc
http:/www.stat.ubc.ca/~bouchard/courses/stat302fa201516/
Intro to Probability
Instructor: Alexandre Bouchard
Fall 2015
Wednesday, September 30, 15
Plan for today:
CORRECTIONS
(#saves + 1) / (#shots + 2)
Calculation for predictive probability: see
p.
1
Math 302 Sample Final
Instructions:
Write each answer very clearly below the corresponding question. Simplify your answer as much as possible but answers may include factorials, choose symbols or the
2
a
exponential function. You may also use the funct
1
Math 302 Sample Final  partial solutions
1. (a) Carefully dene: A, B, C are independent events.
sol. For any pair, as well as for all three, the probability of the intersection is the
product of the probabilities.
(b) Suppose that X, Y and Z are indepe
MATH/ STAT 302
Introduction to Probability
Midterm Exam 1
Section 202
Feb 16, 2007
Time: 50 minutes for working
Name:
Student No:
Instructions:
Read the whole exam paper carefully. We suggest you start with the
questions you think are easiest.
You must ex
Covariance of random variables
Definition
The covariance between X and Y , denoted by Cov(X, Y ), is a measure of
the linearity between X and Y ; it is defined as
STAT 302: Introduction to Probability
Cov(X, Y ) = E [(X
Ed Kroc
E(X)(Y
E(Y )] .
Alternative
Conditional expectation
Discrete case:
Recall that the conditional probability mass function of X given Y = y,
for pY (y) > 0, is:
STAT 302: Introduction to Probability
Pr(X = x, Y = y)
p(x, y)
=
.
Pr(Y = y)
pY (y)
pXY (xy) = Pr(X = xY = y) =
Ed Kroc
T
Moment generating functions (MGFs)
Given a random variable X, we define the moment generating
function of X as follows:
(P
etx pX (x)
(discrete case)
tX
MX (t) = EX (e ) = R 1x tx
 cfw_z
e
f
(x)dx
(continuous case)
X
1
STAT 302: Introduction to Probabil
Chapter 8: Limit Theorems
Learning Outcomes:
Aim: Apply Markovs Inequality, Chebyshevs Inequality, the Weak Law of
Large Numbers, and the Central Limit Theorem in describing the
distribution of and calculating probabilities concerning general random
varia
Weak Law of Large Numbers
Weak Law of Large Numbers
Let X1 , X2 , . . . be a sequence of independent and identically distributed
(i.i.d.) random variables, each having finite mean E(Xi ) = and finite
variance Var(Xi ) = 2 . Then for any " > 0,
X1 + + Xn
l