Probability Cheatsheet v2.0
Compiled by William Chen (http:/wzchen.com) and Joe Blitzstein,
with contributions from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and
Jessy Hwang. Material based on Joe Blitzsteins (@stat110) lectures
(http:/stat110.net) and Blitzs

QUIZ 11: Stab52 - An Introduction to Probability
LAST NAME: FIRST NAME:
STUDENT NUMBER: TUTORIAL:
Problem 1: [4 + 2 = 6 Points] Consider X,- N Bernoulli(t9) , 1} = 1,2, - - -. Furthermore,
X1, X2, - - - are independent. Deﬁne sample mean as following
_

QUIZ 10: Stab52 - An Introduction to Probability
LAST NAME: FIRST NAME:
STUDENT NUMBER: TUTORIAL:
Problem 1: Consider a random variable X follows Binomial(10, 0.6). (a) Using Markov’s
inequality, ﬁnd the upper bound of P (X Z 2). (b) Find the exact numb

UNIVERSITY OF TORONTO AT SCARBOROUGH
FINAL EXAMINATION, AUGUST 2008
STAB52
Duration - 3 hours
AIDS ALLOWED: THIS EXAM IS OPEN BOOK (NOTES)
Calculator (No phone calculators are allowed)
LAST NAME_
FIRST NAME_
STUDENT NUMBER_
This examination consists of tw

UNIVERSITY OF TORONTO SCARBOROUGH
Department of Computer and Mathematical Sciences
Midterm Test, Fall - 2014
STAB52H2: An Introduction to Probability
Duration: One hour and fifty minutes (110 minutes)
LAST NAME:
FIRST NAME:
STUDENT NUMBER:
SIGNATURE:
TUTO

STAB52 Quiz 7
Last Name:
First Name:
Student Number:
1) [5 Marks] Let X and Y be continuous random variables having the following joint density function
f ( x, y) x y for 0 x, y 1
a) Calculate the marginal distribution functions f (x) and f ( y) .
b) Usin

STAB52, An Introduction to Probability (Fall 2014)
Quiz 3, Version 1
1. Let X be a Poisson(2) distribution. Calculate P(X 2). (6 marks)
Solution: Using the probability mass function of the Poisson distribution, we compute
P(X 2) = P(X = 0) + P(X = 1) + P(

Last Name:
First Name:
Student Number:
STAB52 Quiz 9
Question 1 (4 marks)
Given that X ~ Binomial(4,1/2) and Y = 3X+4.
a) Find the Variance of Y (ie, Var(Y) )
b) Find the Covariance of X and Y (ie, Cov(X,Y) )
Solution:
a)
Var(Y) = Var(3X+4) = 9Var(X) = 9*

STAB52, An Introduction to Probability (Fall 2014)
Quiz 2, Version 1
1. Assume that in any family, each child is equally likely to be a boy or a girl. A family with three children
is chosen at random. Find the probability that the family has exact 2 girls

10. Expectation
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Expected Value
RV X measures some aspect of a random experiment
Distribution of X defines uncertainty around X
Often, want to describe aggregate/long-run behavior of X using
single value
Expect

5. Random Variables
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Random Variables
Event (A)
Consider experiment of rolling 2 dice
Sample space |S| = 62 = 36
How do you define an event?
List events sample points
E.g. A = cfw_(3,1),(2,2),(1,3)
Describe t

UTSC
Department of Computer and Mathematical Sciences
STA B52 Term Test
Summer 2014
Student Number:
Family Name:
Special Instructions:
1. You have 120 minutes to attempt 8 equally weighted questions for a possible
200 marks.
2. You must show all your work

6. Discrete Distributions
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Discrete Distributions
Distribution of discrete RV Xcfw_x1,x2, is determined by
collection of all probabilities of the form
P( X xi ) P s : X (s) xi pX ( xi ), i 1, 2,
Viewed as discret

3. Conditional Probability
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Conditional Probability
Imagine I roll a die in the room next door
What is the probability of a 6?
If I told you the roll was even, what is the probability of a 6?
For events A, B th

7. Continuous
Distributions
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Continuous RVs
A RV X is called continuous if P(X=x)=0, x (i.e. PMF=0)
But can P(S)=1 if P(X=x)=0 , x?
Yes, if X takes on uncountably infinite many values.
E.g. can have P(X[a,b])>0

2. Counting
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Discrete Sample Spaces
Sample spaces with distinct outcomes called discrete; can be
Finite: finite # of elements
Ex. Roll of a die, toss of a coin
Countably infinite: elements can be put in 1-to-1

1. Probability Basics
STAB52 F16
Sotirios Damouras
About Me
Sotirios Damouras
Pronounced s-aw-t-EE-r-ee-aw-s or Sam
Background
PhD in Statistics (thesis on Time Series)
Also interested in Mathematical Finance
Contact
Email: sdamouras@utsc.utoronto.

11. Discrete 2D
Distributions
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Multivariate Distribution
Can have multiple RVs defined in random experiment
X 1 (s)
E.g. roll two 6-sided dice and let
X1 = value of 1st die
X2 = value of 2nd die
S
s
X 2 ( s)
D

9. 1D Change of Variable
STAB52 F16
Sotirios Damouras
STAB52 F16
1
Change of RV
Assume RV X follows certain distribution & RV Y=h (X) defined
as some function h of X (k.a. change/transformation or RV)
Y () h( X ()
S
X ()
s
h()
X ( s)
Y (s) h( X (s)
How

Version 1
Last Name:
First Name:
Student Number:
STAB 52 Quiz 8
Question 1 (3 Marks each)
Consider two discrete random variables X and Y. Moreover, X and Y are independent. Given that X and Y
are the following distributions find E(X+Y), E(2X-3Y) and E(4XY

STAB52, An Introduction to Probability (Fall 2014)
Quiz 1, Version 2
1. if a random number is chosen from the set cfw_1, 2, 4, ., 1000, what is the probability that the number
chosen is divisible by 2, 6 or both of them? (6 marks)
Solution: Let An be the

STAB52H: Introduction to Probability
Fall, 2014
Instructor: Jabed Tomal
Department of Computer and Mathematical Sciences
University of Toronto Scarborough
Toronto, ON
Canada
September 4, 2014
Jabed Tomal (U of T)
Probability
September 4, 2014
1 / 45
Defin

STAB52H: Introduction to Probability
Fall, 2014
Instructor: Dr. Jabed Tomal
Department of Computer and Mathematical Sciences
University of Toronto Scarborough
Toronto, ON
Canada
October 1, 2014
Jabed Tomal (U of T)
Probability
October 1, 2014
1 / 52
Conti

STAB52H: Introduction to Probability
Fall, 2014
Instructor: Jabed Tomal
Department of Computer and Mathematical Sciences
University of Toronto Scarborough
Toronto, ON
Canada
October 21, 2014
Jabed Tomal (U of T)
Probability
October 21, 2014
1 / 22
Conditi

STAB52H: Introduction to Probability
Fall, 2014
Instructor: Jabed Tomal
Department of Computer and Mathematical Sciences
University of Toronto Scarborough
Toronto, ON
Canada
September 17, 2014
Jabed Tomal (U of T)
Probability
September 17, 2014
1 / 41
Ran

STAB52H3: Introduction to Probability
Fall, 2014
Instructor: Jabed Tomal
Department of Computer and Mathematical Sciences
University of Toronto Scarborough
Toronto, ON
Canada
November 26, 2014
Jabed Tomal (U of T)
Probability
November 26, 2014
1/9
Stochas