Example of Poisson Distribution-Wars by Year
Number of wars beginning by year for years 1482-1939.
Table of Frequency counts and proportions (458 years):
wars
0
1
2
3
4
More
Frequency Proportion
242
0.5284
148
0.3231
49
0.1070
15
0.0328
4
0.0087
0
0
Tot
Continuous uniform probability model
Math 30530, Fall 2013
August 31, 2013
Math 30530 (Fall 2012)
Continuous models
August 31, 2013
1/2
Continuous uniform probability model
1
Experiment: Have a region R in the plane, and select a point
from it at random (
Miscellaneous discrete random variable examples
Math 30530, Fall 2013
September 27, 2013
Math 30530 (Fall 2012)
Discrete RV examples
September 27, 2013
1/7
Negative Binomial
10% of the population has bloodtype AB-. A researcher samples from
a pool of volu
Bayes Formula and examples
Math 30530, Fall 2013
September 8, 2013
Math 30530 (Fall 2012)
Bayes formula
September 8, 2013
1/7
Example: Who did I beat?
When I play chess, I play Alice (10% of time), Bob (40% of time), and
Carole (50% of time). I beat Alice
The basic rules of counting
Math 30530, Fall 2013
September 13, 2013
Math 30530 (Fall 2012)
Counting
September 13, 2013
1 / 12
Basic counting rule 1 The sum rule
Sum rule 1: if an experiment can proceed in one of two ways,
with
n1 outcomes for the rst way
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 9 not to be turned in
General information
Here are some problems from Section 4.1. This homework isnt to be turned in; it is just
serving as preparation for Fridays exam. I should post
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 7 due in class Wednesday, November 6
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you sho
Math 30530 Introduction to Probability
Quiz 5 Monday December 9, 2013
Solutions
Instructions: This is a closed-book quiz. Please do not use any notes.
1. I toss a fair coin repeatedly until I rst toss a Head. Let X be the number of times I have to toss th
Math 30530 Introduction to Probability
Quiz 3 Wednesday October 9, 2013
Solutions
1. I have ve M&M candies in a bag; three are red and two are green. I take out three at random, and count
how many red ones I selected. Let X be this number. Calculate the e
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 7 Solutions
1. Chapter 2, problems 38, 39, 40 and 41 (a, b, c only) see Homework 7 solutions le
1 on website.
2. By writing the Negative Binomial random variable with parameters p (succ
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 8 Solutions
1. Verify that the variance of the standard normal random variable Z with parameters
2
= 0 and 2 = 1 is indeed 1, by explicitly computing (1/ 2 )x2 ex /2 dx. You
2
can assu
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 9 solutions
1. (a) Let X be an exponential random variable with parameter 1 , and Y be an exponential random variable with parameter 2 . If X and Y are independent, compute
the density
Math 30530 Introduction to Probability
Quiz 2 Wednesday September 25, 2013
Solutions
1. I shue a deck of 52 cards, and then deal them all out one after another. What is the probability that I see
all four aces one after another with no cards in between? (
Math 30530 Introduction to Probability
Quiz 1 Wednesday September 10, 2013
Solutions
1. Use the Probability Axioms to show that for any event A, Pr(Ac ) = 1 Pr(A). Each time you use one
of the axioms, say that you are using it.
Solution: A and Ac are disj
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 6 solutions
1. Let X be a Poisson random variable with parameter . Show that the variance of X
is .
Solution:
2
E (X ) =
k
2
k
k!
k=0
e
= e
(k (k 1) + k )
k=0
= e
(k (k 1) k )
k=0
= e
k
Independence of events (with example)
Math 30530, Fall 2013
September 11, 2013
Math 30530 (Fall 2012)
Independence
September 11, 2013
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Independence of two events
A is independent of B if
Pr(A|B ) = Pr(A)
Math 30530 (Fall 2012)
Independence
September 11
Discrete probability models
Math 30530, Fall 2013
August 28, 2013
Math 30530 (Fall 2012)
Discrete models
August 28, 2013
1/2
Discrete probability models
1
Experiment: nitely many outcomes, or countably (listably) many
Math 30530 (Fall 2012)
Discrete model
Weak Law of Large Numbers & Central Limit
Theorem
Math 30530, Fall 2013
December 10, 2013
Math 30530 (Fall 2013)
Limit Laws
December 10, 2013
1/7
Weak Law of Large Numbers
X1 , X2 , X3 , . . . , Xn are independent copies of the same random
variable, all w
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 1 due in class Friday, September 6
General information
Homework is an essential part of your learning in this course, so please take it very seriously.
It is extremely important that yo
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 10 solutions
1. (a) Let X be a uniformly selected random number on the interval [0, 1]. For a > 0
and b R, let Y = aX + b. Calculate the density function of Y .
Solution: Possible value
Math 30530: Introduction to Probability, Fall 2013
Midterm Exam II
Practice exam
1. Im taking part in the All-Ireland hay-tossing championship next week (hay-tossing is a real
sport in Ireland & Scotland see e.g. http:/scottishheavyathletics.com/sheaf.htm
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 4 due in class Friday, September 27
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you shou
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 2 due in class Friday, September 13
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you shou
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 10 due in class Monday, December 9
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you shoul
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 8 due in class Friday, November 15
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you shoul
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 6 due in class Friday, October 11
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you should
Introduction to Probability, Fall 2013
Math 30530 Section 01
Homework 5 due in class Friday, October 4
General information
At the top of the rst page, write your name, the course number and the assignment number.
If you use more than one page, you should
Math 30530: Introduction to Probability, Fall 2013
Midterm Exam II
Practice exam solutions
1. Im taking part in the All-Ireland hay-tossing championship next week (hay-tossing is a real
sport in Ireland & Scotland see e.g. http:/scottishheavyathletics.com
Examples involving conditional probability
Math 30530, Fall 2013
September 5, 2013
Math 30530 (Fall 2012)
Conditional examples
September 5, 2013
1/5
Im always late to work
60% of days I walk to work. On those days
Im late 80% of the time
Im on time 20%
Discrete Random Variables
Math 30530, Fall 2013
September 18, 2013
Math 30530 (Fall 2012)
Discrete RVs
September 18, 2013
1/2
Setup for discrete random variables
To each outcome of experiment, assign numerical value X ( )
Math 30530 (Fall 2012)
Discrete R