What is Probability?
Quantification of uncertainty. Mathematical model for things that occur randomly. Random not haphazard, don't know what will happen on any one experiment, but has a long run order. The concept of probability is necessary in
STA256H5F
Final Exam Review
(Part I Probability)
Part I Set Operations
The union of A and B (A B) is the set of all points in A or B or both, denoted by
A B
A B=cfw_ x
S: x
A or x
B
Venn Diagram
The intersection of A and B (A B) is the set of all point
STA256H5F Final Review Practice Problems Part II
1. A group of four components is known to contain two defectives. An inspector tests the
components one at a time until the two defectives are located. Once she locates the two
defectives, she stops testing
STA256H5F
Final Exam Review
(Part IV Multivariate Probability
Distributions)
Part I Joint Distribution Function
Definition
Let
Y 1 and Y 2 be discrete random variables. The joint (or bivariate) probability mass
Y 1 and Y 2 is given by
function (pmf) for
p
STA256H5F Final Review Practice Problems Part I
1. Suppose a family contains two children of different ages, and we are interested in the gender of
these children. Let F denote that a child is female and M that the child is male and let a pair such
as F M
Last name (print),
First name (print):
Student #:
UNIVERSITY OF TORONTO
Faculty of Arts and Science
DECEMBER 2013 EXAMINATIONS
STA257H1F, Section L0101
Duration  3 hours
Examination Aids:
NonProgrammable Calculator. One page formula sheet attached (page
Last name (print),
First name (print):
Student #:
UNIVERSITY OF TORONTO
Faculty of Arts and Science
DECEMBER 2011 EXAMINATIONS
STA257H1F
Duration  3 hours
Examination Aids:
NonProgrammable Calculator. No aid sheet allowed.
Show your work in the space pr
Warning (2016): Parts of some problems are not covered for 2016 test, but can apply to
final exam (regarding expectation and variance)
Last name,
First name:
. Student #:
STA257H1 F, Section L0101, Term Test, October 24, 2011
Duration: 110 min. Allowed: h
STA256H5F
Final Exam Review
(Part II Discrete Random Variables)
Part I Discrete Random Variables ()
Definition
A random variable Y is said to be discrete if it assumes only a finite or countably infinite number
of distinct values.
1
Part II Discrete Pr
STA256H5F Final Review Practice Problems Part III
Y
1. A random variable
has the following distribution function:
cfw_
0, for y <2,
1/8, for 2 y <2.5,
3 /16, for 2.5 y <5,
F ( y )=P ( Y y ) = 1/2, for 4 y <5.5,
5 /8, for 5.5 y <6,
11/16, for 6 y <7,
1, fo
STA256H5F Final Review Practice Problems Part IV
1. Let
Y 1 and Y 2 have the joint probability density function given by
cfw_
f ( y 1 , y 2) = k ( 1 y 2 ) , 0 y 1 y 2 1,
0,elsewhere .
k that makes this a probability density function.
b) Find P(Y 1 3/ 4,Y
Last name,
First name:
. Student #:
STA257H1 F, Section L0101, Term Test, October 24, 2011
Duration: 110 min. Allowed: handcalculator. Show your work in the space provided.
Aid sheet attached (last page). You may use page backs for preparations.
In every
f
F
r
g
D
f
ipc AAt v @8 c FD Y P vX t
i % t Y Deg f6X u
ic S S d t e'Hq v B ud e8 s D Aeg Q g Y AeD 6X x Y
upv t hd A8 s d t s AD f y
TX 8 Bw 5 z y Q ~B i p3C c 5 cfw_a 6 WV 4 iWT a tx XR T lV ds vwER t vWUS uXT ts a d6 R r s i 8 wt7 I 0a 97 pc qA8 T
B
q g y @ wi8i g q w q i y q 7m7h7Q9q9xfmmV w y q q w vi u yi y y w @ q r q u q p w 6 q W y q pi w q q w q y y w yi vi q W y w vi g P mo7dtQke7m7YrQmxh7s9cpmrAxYhxrmQxiAQ9cy!AQQ y8i W v y q y q p w 6 @8 q i8 @88 6 q w q q q w F y y fyA7ph7h7kQmxhQAm7A7!
STA257: Probability and
Statistics I
Section L0101 Fall 2015
Week 3, Lecture 6
Discrete Random Variables (2.1)
1
Discrete Probability
Distributions
List of common discrete probability
distributions:
Bernoulli
Binomial
Geometric
Negative Binomial
Hypergeom
STA257: Probability and
Statistics I
Section L0101 Fall 2015I
Week 2, Lecture 4
1
Partitions
Definition: A partition of is a finite collection of
events cfw_B1,B2,.,Bn, such that
Bi
Bj
n
, i
B2
j
A
B1
UB
i
B3
i
For any event A, and partition cfw_B1,B2,.,B
STA257
PROBABILITY
Ch 3, Joint Distributions, Part 2
Instructor
Dragan Banjevic
Fall 2016
1
Independent Random Variables (I)
(Why in red? Because the independence is very important, it is
one of the key notions in Probability. Then, pay attention!)
Whe kn
STA257
PROBABILITY
Ch 5, 6 Limit Theorems
Instructor
Dragan Banjevic
Fall 2016
1
The Law of Large Numbers (I)
We built our theory hoping to get a good model for real
phenomena of random experiment, random events, and their
probabilities. Our main interpre
STA256H5F
Final Exam Review
(Part III Continuous Random Variables)
Part I Continuous Random Variables
Continuous Random Variables

A random variable
Y
is called continuous if it can take any value within a finite or infinite
interval of the real line.

STA257 Probability and Statistics I
Instructor
Email
Office Hours
Office
Course Website
Github
Neil Montgomery
[email protected]
Any time
BA8137
http:/portal.utoronto.ca
https:/github.com/sta257fall2016
General Information
The course website will b
2011 STA257 Term Test
(a) [2] The adult needs glasses.
(b) [2] The adult needs glasses but does not use them.
(c) [2] The adult uses glasses whether the glasses are needed or not.
(d) [3] The adult uses glasses, if the glasses are needed.
(e) [4] The adul
STA 257H1 F  PROBABILITY AND STATISTICS I
Summer 2016 (May 9 to June 24)
Lectures:
Instructor:
Mondays and Wednesdays 710pm in MP 203
Dr. Shivon SueChee (Email: [email protected])
Office hours: Mondays and Wednesdays 56pm in SS 6026
Course
Ch2 Homework Solution
28 Show that the binomial probabilities sum to 1.
X ~ B (n, p)
n
n
k pk (1 p)nk = ( p + (1 p)n = 1
k =0
211 Consider the binomial distribution with n trails and probability
p of success on each trial. For what value of k is P(X=k
STATISTICAL LABORATORY, March 29th, 2011
UNIVARIATE PROBABILITY DISTRIBUTIONS
Manuela Cattelan
1
GENERAL CONTINUOUS DISTRIBUTIONS
Ex1 A line segment of length 1 is cut once at random. What is probability that the
longer piece is more than twice the length
FSRM 582, Homework 3  Solutions
Assigned: September 16, 2014
Due: September 30, 2014
Problem 1. Suppose A1 and A2 are independent events for some probability measure P on .
For j = 1, 2, define
1,
! 2 Aj
Ij (!) =
0, otherwise,
to be the indicator functio
STATISTICAL LABORATORY, March 22nd, 2011
CONDITIONAL PROBABILITY AND
INDEPENDENCE
Manuela Cattelan
1
SIMPLE PROBLEMS
Ex1 A couple has two children. 1) What is the probability that both are girls, given that
the oldest is a girl? 2) What is the probability
Chapter
1.7 Prove Bonferronis inequality :
P( A B) P(A) + P(B) 1
SOL :
P(A B) = P(A) + P(B) P(A B) 1
P( A B) P(A) + P(B) 1
1.8 Prove that
n
n
P( ! Ai )
P( A )
i
i =1
i =1
SOL :
n
prove that P( ! Ai )
i =1
n
P( A )
i
i =1
2
n = 2 P( ! Ai ) = P(A 1 A 2
Convolution de variables alatoires continues et applications
Motivation de la dfinition.
Si X et Y sont deux variables alatoires continues indpendantes de densits respectives et , la
cumulative H de est donne par :
o F est la c
STA257
PROBABILITY
Ch 2, Random variables, Part 1
Instructor
Dragan Banjevic
Fall 2016
1
Ch. 2 Random Variables
Remember experiment: Observing weather today
Conditions
Outcomes temperature, pressure, precipitation,
wind, humidity, etc.
Events . , It w
What Have We Learned?
Professor Malinova, ECO358
October 04
term test 1 info.
Test Logistics
When?
35pm (start at 3:10pm sharp) for L0101
57pm (start at 5:10pm sharp) for L5101
Schedule conflicts, etc.?
Will try to accommodate but email Professor
STA257
PROBABILITY
Ch 2, Random variables, Part 2
Instructor
Dragan Banjevic
Fall 2016
1
Ch. 2 Continuous Variables
Remember experiment: Observing weather today
Outcomes temperature, pressure, precipitation,
wind, humidity, etc.
Events . , It will be ho