Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Lectures 1221
Random Variables
Denition: A random variable (rv or RV) is a real valued function dened on the sample space.
The term random variable is a misnomer, in view of the normal usage of function and variable.
Random variables are denoted by capita
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
F.W. Scholz
Nov. 5, 1993
Math/Stat 394 Midterm Solutions
1) E F c means that E is outside F and thus E F = . Thus P (E c F c ) = 1 P (E F ) =
1 P (E) P (F ) a).
2) E c F implies that F covers the outside of E, E F = S d).
3) The number of distinct words i
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Your Name
F.W. Scholz
Feb. 11, 2011
Math/Stat 394 A Midterm
Closed Book! One Crib Sheet
For the rst ve problems only circle the unique correct answer.
For problems 68 show your work!
1. (7 points) For any events E and F with E F c we also have:
a) F E =
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Lecture 2227
Continuous Random Variables
So far we have dealt only with discrete RVs, with nite or countably innite value sets. Now we
want to extend that notion to RVs with much wider value sets (e.g., the continuum of intervals).
Denition: We call a ran
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Math/Stat 394, Winter 2011
F.W. Scholz
Central Limit Theorems and Proofs
The following gives a selfcontained treatment of the central limit theorem (CLT). It is based on
Lindebergs (1922) method. To state the CLT which we shall prove, we introduce the fo
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Lectures 15
Probability Models
Analogy with Geometry: abstract model for chance phenomena
Language and Symbols of Chance Experiments:
Sample space S, consisting of all possible outcomes (elements) e, f, . . .,
events (subsets of S) E, F, . . . E occurs wh
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Math/Stat 394 A
F.W. Scholz
The Two Pair Poker Problem
Believe it or not?
In poker, dont forget, if youre dealt a pair, that automatically increases the odds that your
opponent has been dealt a pair, too. Mike Mailway, Post Intelligencer, 9/26/1992.
Let u
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
F.W. Scholz
Nov. 3, 1993
Math/Stat 394 Midterm
Closed Book! For the rst ve problems only circle the correct answer.
For problems 6, 7 and 8 show your work!
1) (5 points) For any E F c we have that:
a) 1 P (F ) P (E),
b) 1 P (E),
c) 1,
2) (5 points) E c F
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
F.W. Scholz
March 14, 2011
Math/Stat 394 A Final & Solutions, Winter 2011
Closed Book! One 2Sided Crib Sheet
Instructions: For the multiple choice questions (110) only circle the correct item number,
no work needs to be shown. For the remaining problems
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Math/Stat 394
F.W. Scholz
PoissonBinomial Approximation
Theorem 1: Let X1 and X2 be independent Poisson random variables with respective parameters
1 > 0 and 2 > 0. Then S = X1 + X2 is a Poisson random variable with parameter 1 + 2 .
Proof:
P (X1 + X2 =
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Solutions to Math/Stat 394 Final, Fall 1993
1. = 1/100 P (N (50) = 0) = exp(50/100) = .6065 closest to d).
2. P (10 Ei ) = 1 P (10 Ei )c ) = 1 P (10 Eic ) = 1 .910 = .6513, closest
i=1
i=1
i=1
to f).
3. Let Hi (Ti ) be the event of exactly i heads (tails)
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Stat 394 A: Homework 4
1. From the basic denition of the conditional probability P (EF ) = P (EF )/P (F ) for P (F ) > 0 we
had obtained the product formula P (EF ) = P (EF )P (F ) for two events E and F . Using this two
event product formula prove the
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
Math/Stat 394
Fall 1993
F.W. Scholz
Final
Instructions: In the multiple choice questions (110) circle only the correct item number.
In the remaining problems show all your work.
The test has four pages plus a normal table.
Closed book. One twosided sheet
Introduction to Probability Theory and its Applications
STAT 394A

Spring 2011
The BoyGirl Conundrum
A well known problem in elementary probability theory is as follows. Suppose we have a family
with two children, one of which is known to be a boy. What it the chance that the other is a boy
as well?
In order to solve this, certain