Sta347 Probability I
Selected Practice Problems for Final
Dec. 09, 2014
(1) Let X have the density function f (x) = cxe2x , 0 x < . And f (x) = 0 otherwise.
(a). Find the value of c.
(b). Give the mean and the variance of X.
(c). Give the moment generatin
STA347
Test2
Time:2hrs
Note: Each question is worth 10 marks. No aids are allowed. Some information
may be found at the end.
1. Let cfw_N(t):t0 be a Poisson process of rate 2. Suppose you know N(1)=5
and those 5 points occur at times T 1 <.<T 5 . Let Y=T
APM236HW4due April 3
NAME:
1. Consider the following information about the nal tableau of LPP:
0 21 1
4
1
2 0 1
1
0
2 1 0 2 x B = 2 c =
T = 1
2
1
0
2 0 1 21 1
3
1
3
2
1
1
3
2
a) As part of a marketing campaign we are asked to reduce the price of item 6
APM236HW3due March 13
NAME:
1. consider question 21 of section 2.1, and see the answer at the back. Use this information to answer
the following questions in the order that they appear. You must use the information obtained in the
previous parts of the
APM236HW1due Jan. 23

NAME:
1. Read section 0.1 and do the following questions (these are very important throughout the course:)
a) 4(d)
b) Read and understand formula (2) page 3, and use it to do exercise 13(a,b)
Page 1 of 8
APM236HW1due Jan. 23
c)
Sta347H1 F 2016 Course Information
This course is an introduction to probability from a nonmeasure theoretic point of
view. Random variables/vectors; independence, conditional expectation/probability and
consequences. Various types of convergence leading
Sta347 Probability I
Homework 1
Sep. 25, 2016
Due Oct. 4, 2016 in class
No late Homework will be accepted.
(1) Problems 4 and 5 on Page 8 of the Textbook. (Note that there is a typo in Problem
4. A(X)2 should be changed to A(X 2 ) in the first equation i
Sta347 Probability I
Homework 4
Nov. 24, 2016
Due Dec. 6, 2016 in class
You should work out this Homework individually. Group works or discussions are not
acceptable.
No late Homework will be accepted.
(1) Problem 4 on Page 144 of the Textbook.
(2) Prob
Sta347 Probability I
Selected Practice Problems for Final
Dec. 08, 2016
(1) Let X have the density function f (x) = cxe2x , 0 x < . And f (x) = 0 otherwise.
(a). Find the value of c.
(b). Give the mean and the variance of X.
(c). Give the moment generatin
Sta347 Probability I
Homework 2
Oct. 25, 2016
Due Nov. 1, 2016 in class
No late Homework will be accepted.
(1) Problem 7 on Page 45 of the Textbook.
(2) Problem 1 on Page 65 of the Textbook.
(3) Problem 6 on Page 65 of the Textbook.
(4) Customers leaving
Solution to Homework #3
prepared by Zhenhua Lin
November 29, 2016
1. Let A denote the event that the first player holds all four aces, and B the event that the first player holds the
ace of hearts. Note that A B, which implies that A B = A. To compute Pr(
STA 347 Homework 4 Solution
David Farahany, Luhui Gan
December 11, 2016
Problem 1. Buffons needle problem (Problem 4 on Page 144 of the Textbook). Show that
if L < D, then the probability that the needle intersects a crack between two floorboards is
2L
.
APM236HW4due April 9

NAME:
1. Consider a transportation problem with supply and demand vectors S = [50, 60] and d = [20, 50, 40]
and the cost matrix
4 3 5
C=
3 6 2
a) Use min cost method to nd an initial solution (and set of basic variables for this p
APM236HW2due Feb. 13
NAME:
1. Carefully read section 2.1 and answer the following questions. Your answers must be illustrated algebraically.
a) Read the optimality condition presented in page 108 and explain why this condition guarantees
that the soluti
STA347 Probability
Gun Ho Jang
July 11, 2016
Note: This note is prepared for STA347. There might be numerous fault arguments/statements/typos. If
you spot one, please contact the instructor or you may look up references which may contain errors too.
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HW 3 Solutions
Jonathan Auerbach
October 1, 2014
1. Problem 4.4 #4
Suppose that X is a random variable such that E(X 2 ) is finite. (a) Show that E(X 2 ) [E(X)]2 . (b) Show
that E(X 2 ) = [E(X)]2 if and only if there exists a constant c such that P r(X =
STA347  Probability I  MidTerm Test  2015 Summer Solution
1.
(a) (i) P (A) 0 for any A F, (ii) for any disjoint sets A1 , A2 , . . . F, P (n=1 An ) =
n=1 P (An ), (iii) P () = 1.
(b) satisfies (i), (iii) and finite additivity along with continuity fro
STA347 Problem Set 2
Problem 1. Suppose P (AB) = P (A)+P (B) if A, B are disjoint. Show that, for a disjoint sets A1 , . . . , An ,
P(
n
[
Aj ) =
n
X
P (Aj ).
j=1
j=1
Problem 2. Suppose a measure P satisfies P () < , P (A) 0 for all A F, for any disjoint
STA347 Problem set #1
Problem 1. Using the definition of probability measure, prove that
P (A B) = P (A) + P (B) P (A B).
Problem 2. Let An be a sequence of events. Prove Booles inequality, that is,
P(
An )
n=1
P (An ).
n=1
Problem 3. An is a monotone de
STA347 Problem set #1
Problem 1. Using the definition of probability measure, prove that
P (A B) = P (A) + P (B) P (A B).
Solution. Theorem 1 in lecture note.
Problem 2. Let An be a sequence of events. Prove Booles inequality, that is,
P(
An )
n=1
P (An
UNIVERSITY OF TORONTO
Faculty of Arts and Science
AUGUST 2014 EXAMINATIONS
STA347H1S
Duration  3 hours
Examination Aids: None
Last name:
First name:
Student number:
Middle name:
Instruction
1. If a question asks you to so some calculations or derivations
STA347

Probability
I

Quiz

2015
Summer
1. T, F, T, F, F, F, T, F, F, F, T, F, F, F, T, T, T, T, F, F
R 3
RRy
5
5
y y dy = c()
2. (a) 1 = 0 0 c()x2 y dx dy = c()
3
15 . Hence c() = 15/ .
0
(b) The marginal density of Y is, for 0 < y < ,
Z y
pdfY (y) =
STA347  Probability I  MidTerm Test  2014 Summer
Last Name:
Student Number:
First Name and Initials:
No Aids are Allowed
Instruction
1. Dont forget writing your name and student number.
2. If you do not have enough space, use the other side and refer
STA347

Probability
I

MidTerm
Test

2014
Summer
Solution (Problem 1). (a) C  N = n Binomial(n, 1/2).
(b) If U Poisson() and V  U = n Binomial(n, p), then V Poisson(p) because
P (V = k) =
P (V = k, U = n) =
n=k
P (U = n)P (V = k  U = n) =
n=k
n=k
e
Math 151. Rumbos
Spring 2008
1
Solutions to Assignment #20
1. Let X1 , X2 , X3 , . . . denote a sequence of independent, identically distributed
random variables with mean . Assume that the moment generating function
of X1 exists in some interval around 0
UNIVERSITY OF TORONTO
Faculty of Arts and Science
December Final Examination, 2013
STA347H1F
Duration 2 hours
Examination Aids: NonProgrammable Calculators. One A4 Size
Handwritten Formula Sheet
Name (Print Clearly!):_._ Student ID:
Do not turn the page
UNIVERSITY OF TORONTO
Faculty of Arts and Science
December Final Examination, 2014
STA347H1F
Duration 2 hours
Examination Aids: NonProgrammable Calculators. One A4 Size
Handwritten Doublesided Formula Sheets
Name (Print Clearly!):_ Student ID:
Do not tur
Categorical Data Analysis
(Chapter 13)
Note: This material is not allowed to be published or posted without the
written permission of the author and textbook publisher.
Contents
1. Introduction . 2
2. The
test for independence . 3
2.1 Contingency table an
Descriptive Statistics
and
Introduction to R
(Chapter 10)
Note: This material is not allowed to be published or posted without
the written permission of the author and any third parties.
1. The empirical cumulative distribution function and the survival
f
Inference from Two Samples
(Chapter 11)
Note: This material is not allowed to be published or posted without the
written permission of the author and textbook publisher.
1. Introduction
2. Two independent sample inference for
2.1 Pooled t inference for i
Analysis of Variance (ANOVA)
(Chapter 12)
Note: This material is not allowed to be published or posted without a
written permission of the author and publishers.
Contents
1. Introduction . 2
2. Completely randomized design and oneway ANOVA . 6
3. Post ho
Summary and Examples of Confidence Intervals
The Central Limit Theorem for sample proportion
The sampling distribution of the sample proportion p has approximately a
pq
normal distribution with mean p = p and standard deviation p = ,
n
denoted as
pq
) app