Lecture 17: Important Discrete Distributions,
Expectation, and Variance
1
Homework
Let n be a positive integer. Let S be the set of integers between 1 and n.
Consider the following process: We remove a number from S at random and
write it down. We repeat
STAT 241 Lecture 5
Random variable. Expectation. Probability mass functions.
1/18
Random variable
Definition
A random variable X is a real-valued function defined on sample space:
X : R.
2/18
Random variable
Definition
A random variable X is a real-valued
Fall 2016
STAT 241: Probability Theory with Applications
Homework 5
Due: Oct 7, 2016 in class
Prof. Yihong Wu
Reading: Grinstead-Snell, Sec 5.1 (on Poisson distribution), 12.2
1. Grinstead-Snell: Section 5.1, Problem 24
2. Toss a fair die for 100 times in
STAT 241 Lecture 3
Conditional probability. Independence of events.
1/24
Last time: Probability axioms
Sample space:
Outcome:
Events: A
How to make new events: set operations
2/24
Probability axioms
Definition
1
Positivity: For any event A,
P (A) 0
STAT 241 Lecture 10
Random walk: Gamblers ruin
1/27
Random walk
A particle starts at 0, and at each step it either moves 1 unit to the right
with probability p or to the left with probability q = 1 p, independently.
q
4 3 2 1
p
0
1
2
3
4
Let Sn be the par
STAT 241 Lecture 2
Probability axioms.
1/18
Axiomatic framework
Three elements
1
Sample space
2
Events
3
Probability measure
2/18
Sample space and events
Sample space : set of all possible outcomes
Outcome : element of sample space
Events A : subset of
STAT 241 Lecture 8
Distributions related to independent trials: Bern, Bin, Geo
1/28
Midterm
Davies Auditorium, Oct 14 Friday 9am - 1015am
One letter-size cheatsheet allowd.
Electronic devices NOT allowed
If you have a conflict, please let me know asap
STAT 241 Lecture 6
Functions of random variables. LOTUS rule. Independence of random
variables.
1/22
Functions of random variables
Let X be a discrete random variable. Let f : R R be a function. Then
Y = g(X) is also a discrete random variable,
2/22
Funct
Fall 2016
STAT 241: Probability Theory with Applications
Homework 2
Due: Sep 16, 2016 in class
Prof. Yihong Wu
1. Assume that P (A) > 0 and P (B) > 0. Consider the following statements. If you agree, prove
it; otherwise, present a counterexample.
(a) If P
STAT 241/541
Probability Theory with Applications
Fall 2016
1
Lecture 1: Introduction
2
Instructor
Prof. Yihong Wu, yihong.wu@yale.edu
Office hours
I
I
I
Wed 4-6pm or by appointments
Room 107, 24 Hillhouse Ave (James Dwight Dana House).
First day: this a
Fall 2016
STAT 241: Probability Theory with Applications
Homework 4
Due: Sep 30, 2016 in class
Prof. Yihong Wu
Reading: Grinstead-Snell, Sec 3.2, 4.3, 5.1 (on geometric distribution).
1. Grinstead-Snell: Sec 3.2, Problem 18. Do the following:
(a) Answer i
STAT 241 Lecture 7
Union bound. Include-Exclusion principles.
1/21
Recall: Union of two events
From axioms of probability:
P (A B) = P (A) + P (B) P (AB)
(Again: we omit and write AB for intersection)
2/21
Recall: Union of three events
P (A B C) = P (A) +
STAT 241 Lecture 4
Law of total probability. Bayes formula.
1/28
Example
A statistics professor had to take a flight for the first time and was
terrified of the prospect that someone will bring a knife on the plane.
2/28
Example
A statistics professor h
Homework 6
Problem 1. For 1-dimensional random walk with p 6= 1=2, show that the
probability that the walk never returns to 0 is positive.
Problem 2. Let (i; j) be integers such that i j is even. Show that the
two-dimensional random walk passes through (i
STAT 241 Lecture 9
Poisson distribution
1/25
Poisson distribution
A random variable X is said to have a Poisson distribution with
parameter ( 0), denoted by X Poi(), if1
P (X = k) =
e k
,
k!
k = 0, 1, 2, . . .
i.e., P (X = 0) = e , P (X = 1) = e , etc.
0.