Lecture 2:
Probabilities
Dan Sloughter
Furman University
Mathematics 37
January 6, 2004
2.1
Probability measures
Denition 2.1. Let P be a real-valued function dened on the events of a
sample space . We call P a probability measure if
1. for any event E, 0
Lecture 16:
Sums of Independent Random Variables
Dan Sloughter
Furman University
Mathematics 37
February 9, 2004
16.1
Discrete variables
Theorem 16.1. Suppose X and Y are independent discrete random variables with probability functions pX and pY . Then Z
Lecture 13:
Joint Distributions
Dan Sloughter
Furman University
Mathematics 37
February 2, 2004
13.1
Discrete distributions
Denition 13.1. If X and Y are both discrete random variables, then we
call the function p : R2 R dened by
p(x, y) = P (X = x, Y = y
Lecture 15:
Independent Random Variables
Dan Sloughter
Furman University
Mathematics 37
February 5, 2004
15.1
Independence: discrete variables
Denition 15.1. We say random variables X1 , X2 , . . . , Xn are independent
if for any sets A1 , A2 , . . . , An
Lecture 11:
Continuous Random Variables
Dan Sloughter
Furman University
Mathematics 37
January 29, 2004
11.1
Continuous random variables
Denition 11.1. Suppose X is a random variable and f : R R is such
that for any real numbers a < b,
b
P (a X b) =
f (x)
Lecture 9:
Bayes Theorem
Dan Sloughter
Furman University
Mathematics 37
January 20, 2004
9.1
Bayes theorem
The following result is known as Bayes theorem.
Theorem 9.1. Suppose B1 , B2 , . . . , Bn is a partition of a sample space
and A is an event in . T
Lecture 8:
Partitions
Dan Sloughter
Furman University
Mathematics 37
January 16, 2004
8.1
Partitions
Denition 8.1. If B1 , B2 , . . . , Bn are disjoint events in a sample space
and n Bi = , then we call B1 , B2 , . . . , Bn a partition of .
i=1
Theorem 8
Lecture 7:
Conditional Probability
Dan Sloughter
Furman University
Mathematics 37
January 15, 2004
7.1
Conditional probability
Example 7.1. Suppose two dice are rolled. If is the sample space, A is
the event that the rst die is a 6, and B is the event tha
Lecture 6:
Independence
Dan Sloughter
Furman University
Mathematics 37
January 14, 2004
6.1
Independent events
Example 6.1. Consider the experiment of tossing a fair coin twice and let
A be the event the rst toss is a head and let B be the event the secon
Lecture 4:
Examples with Counting
Dan Sloughter
Furman University
Mathematics 37
January 9, 2004
4.1
Urn models
Example 4.1. Four balls are drawn, without replacement, from an urn with
8 red balls and 5 white balls. If E is the event that the sample has 2
Lecture 1:
Sample Spaces
Dan Sloughter
Furman University
Mathematics 37
January 6, 2004
1.1
The notion of probability
The outcomes of many experiments depend on chance. For example, the
result of rolling a die or ipping a coin, the number of hits per day
Lecture 5: Repeated Experiments
Dan Sloughter Furman University Mathematics 37 January 18, 2004
5.1
Sampling without replacement
Example 5.1. Suppose an urn has 8 red balls and 5 white balls and we sample 4 balls, with replacement. If A is the event that
Lecture 3:
Elements of Counting
Dan Sloughter
Furman University
Mathematics 37
January 8, 2004
3.1
Basic principles and notation
The basic principle of counting: Suppose is the sample space for an experiment which may be performed in k stages, with n1 way
Lecture 10:
Discrete Random Variables
Dan Sloughter
Furman University
Mathematics 37
January 21, 2004
10.1
Discrete random variables
Denition 10.1. If is a sample space and X : R is a function for
which the set cfw_ : X() a is an event in for any a R, the