9. Expectation
STAB52 F17
Sotirios Damouras
STAB52 F17
1
Expected Value
RV X measures some aspect of a random experiment
Distribution of X defines uncertainty around X
Often, want to describe aggre
8. Important Distributions
STAB52 F17
Sotirios Damouras
STAB52 F17
1
Distributions
Look at some additional common distributions:
Discrete
Negative Binomial
Poisson
Continuous
Gamma
Normal
STAB52
7. Continuous
Distributions
STAB52 F17
Sotirios Damouras
STAB52 F17
1
Continuous RVs
A RV X is called continuous if P(X=x)=0, x (i.e. PMF=0)
But can P(S)=1, when P(X=x)=0 , x?
YES, if X takes on un
6. Discrete Distributions
STAB52 F17
Sotirios Damouras
STAB52 F17
1
Discrete Distributions
Distribution of discrete RV Xcfw_x1,x2, is determined by
collection of all probabilities of the form
P( X= x
CSC A67/MAT A67 - Discrete Mathematics, Fall 2017
Assignment #1: Proofs
Due: October 29, 2017 at 11:59 p.m. This assignment is worth 10% of your final grade.
Warning: Your electronic submission on Mar
UNIVERSITY OF TORONTO SCARBOROUGH
Department of Computer and Mathematical Sciences
Midterm Test, June 2017
STAB52 Introduction to Probability
Duration: One hour and fifty minutes
Last Name:
First Name
UNIVERSITY OF TORONTO SCARBOROUGH
Department of Computer and Mathematical Sciences
STAB52H3 Introduction to Probability Summer 2017
Course Description: STAB52 is a mathematical treatment of probabilit
LOCATION 1
POINT 5
BRIEF You are on the First floor of the UTSC AC building.
LONG You are on the First floor of the UTSC AC building. The female washroom is located on (3,3) on the First floor. There
Enhancements The game takes place in the UTSC campus. Want to win the game, the player has to bring all the required items to the exam room on time and have visited the washroom before the exam. One o
Means of expression
Marcotext
-The target text, in this case is a text about a text, or about one or more particular aspects of a
text
- there is 2 types of translation: documental and instrumental
STAB52 F16 - LEC02
Quiz 10
Name (f,l):
ID #:
1. Let the MGF of the RV X be mX (t) = eat , t R and some a R.
i. Find the mean of X.
ii. Find the variance of X.
iii. Show that X is a constant RV with va
STAB52 F16 - LEC02
Quiz 9
Name (f,l):
ID #:
1. Assume you roll 5 fair 6-sided dice; Find the probability that:
i. The minimum roll is equal to 1.
ii. The minimum roll is equal to 3.
Solution: Let X1 ,
STAB52 F16 - LEC02
Quiz 8
Name (f,l):
ID #:
1. Consider rolling two standard fair dice, and let RVs X1 , X2 represent the result of each
die. Define the RVs Z = X1 + X2 and W = |X1 X2 |. Find the cond
UTSC
STAB52H3
MIDTERM EXAM
STUDY GUIDE
find more resources at oneclass.com
Scanned by CamScanner
find more resources at oneclass.com
find more resources at oneclass.com
Scanned by CamScanner
find more
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
STAB52H3 An introduction to Probability
Midterm Examination
October 31, 2016
Duration: 110 minutes
Examination aids all
Probability and Statistics
The Science of Uncertainty
Second Edition
Michael J. Evans and Jeffrey S. Rosenthal
University of Toronto
W. H. Freeman and Company
New York
Senior Publisher: Craig Bleyer
P
UNIVERSITY OF TORONTO SCARBOROUGH
Department of Computer and Mathematical Sciences
Midterm Test, Winter - 2017
STAB57H3: An Introduction to Statistics
Duration: Two hours (120 minutes)
LAST NAME:
FIRS
UNIVERSITY OF TORONTO SCARBOROUGH
Department of Computer and Mathematical Sciences
Midterm Test, Winter - 2017
STAB57H3: An Introduction to Statistics
Duration: Two hours (120 minutes)
LAST NAME:
FIRS
STAB22
Introduction to Statistics
Asal Aslemand (section 1) Ken Butler (section 2)
Srishta Chopra (section 3)
Fall semester 2014
1 / 21
What Statistics is
Data: collecting, organizing, interpreting.
U
Problem Set 19. Moment Generating Functions
1. Find the MGF of the Bernoulli(p) and the Binomial(n, p) distributions. Verify that the Binomial is the sum of n i.i.d. Bernoulli(p) RVs using the MGF met
Problem Set 10. Expectations
1. Two teams reach a best-of-seven playoff, i.e. the first team to win 4 games become champions
and no more games are played beyond that. Assume both teams have the same d
Problem Set 9. 1D Change of Variables
1. Let X Poisson() and Y be the indicator of X being odd. Find the PMF of Y .
(Hint: Find P (Y = 0) P (Y = 1) by writing P (Y = 0) and P (Y = 1) as series and the
Problem Set 11. Discrete 2D Distributions
1. Suppose a fair 6-sided die is rolled twice, and the RVs (X1 , X2 ) represent the values of the first
and second rolls. Let X = X1 + X2 be their sum and Y =
Problem Set 8. Important Distributions
1. Let X NegBinom(r, p) and Y NegBinom(s, p), where X and Y are defined on separate
and independent Bernoulli(p) trial sequences. If Z = X + Y , what is the dist
Problem Set 12. 2D Integrals
1.
i. Set up a double integral of f (x, y) over the part of the unit square, 0 x 1, 0 y 1,
on which y x/2.
ii. Set up a double integral of f (x, y) over the part of the un
def insert(listA, listB, index):
'(list,list,int) -> list
return a copy with listA with listB inserted at the index given
REQ: index <= len(listA)
>insert([1, 2, 3], ['a', 'b', 'c'], 2)
[1, 2, 'a', 'b
def copy_me(lista):
'(list) -> list
Return the copy of the input list with strings in the list become
uppercased, integers and floats have their value plus one, booleans
negated, Lists replaced with l
def percent_to_gpv(mark):
'(int) -> (float)
Return the gpa for a course given the raw percentage mark
REQ: 0 <= mark <= 100
>percent_to_gpv(86)
4.0
>percent_to_gpv(75)
3.0
>percent_to_gpv(48)
0.0
'
#d