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
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
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
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
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
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
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
STA347 Probability
Gun Ho Jang
STA347 Probability () 1
Review of Required Mathematics
Background
STA347 Probability () 2
Set Theory I
A set is a collection of distinct objects.
A member a in a set A is called an element, denoted by a A
or A a.
N = cfw_1,
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

Probability
Last Name:
Student Number:
I

Quiz

2015
Summer
First Name and Initials:
A nonprogrammable calculator is allowed
Instruction
1. Dont forget writing your name and student number.
2. If you do not have enough space, use the other sid
STA347 Problem Set
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
X
P (An ).
n=1
Problem 3. An is a monotone d
STA347 Probability
Gun Ho Jang
STA347 Probability () 1
Chapter 1
Introduction to Probability
Section 1.1
The History of Probability
STA347 Probability () 17
History of Probability I
From the textbook
The concepts of chance and uncertainty are as old as
ci
STA347 Probability
Gun Ho Jang
July 14, 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.
Expec
STA347 Probability
Gun Ho Jang
July 19, 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.
Inequ
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.
Revie
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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
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)
Test
STA347
Time:3hrs
Instructions: The test is out of 100 and each question is worth 7. Your maximum grade
is 100. See the end for some useful information. Please, at most 1 question/page in your
booklets! No aids allowed.
1. Let X be a rv in cfw_0, 1, 2
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
.
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(
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
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 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