6.2 One Function of Two Random Variables
Z g ( X ,Y )
FZ ( z ) Pcfw_g ( X , Y ) z Pcfw_( X , Y ) Dz
f ( x, y )dxdy
(8-1)
Dz
y
Dz
Dz
x
g ( x, y ) z
(boundaries)
1
y
y g ( x)
1
f ( x, y )dxdy a g ( x ) f ( x, y )dy dx
b
Dz
2
Dz
y g ( x)
2
a
g1 ( x )
x
b

Chapter 5: Functions of One Random Variable
5.1. The Random Variable g ( X )
Let X be a r.v defined on the model ( S , F , P ), and suppose g(x) is a
function of the variable x. Define
Y g ( X ).
Is Y necessarily a r.v? If so what is its PDF FY ( y ), pdf

Chapter 6: Two Random Variables
Suppose A and B are two events. We know that in order to study A
and B, just knowing P ( A) and P ( B ) is not enough. We have to know
how they are related to each other. That is we have to know P ( AB).
Similarly, suppose

5.3. Mean and Variance
Example
Two shooters, their shooting techniques are expressed in the
following tables.
Shooter B
Shooter A
10
9
8
points
8
9
10
points
8
9
10
probs
0.3
0.1
0.6
probs
0.2
0.5
0.3
Question: Whose technique is better?
Solution
The answ

Chapter 4 The Concept of a Random Variable
4.1 Introduction
Let (S, F, P) be a probability model for an experiment, and X a function
that maps every S , to a unique point x R, the set of real numbers.
Since the outcome is not certain, so is the value X( )

Chapter 1: The Meaning of Probability
Basics
Probability theory deals with the study of random phenomena, which
under repeated experiments yield different outcomes that have certain
underlying patterns. The notion of an experiment assumes a set of
repeata

4.5 Asymptotic Approximations for Binomial Random Variable
A
A
P ( A) p, P ( A) q, p q 1
n
Pn (k ) Pcfw_ A occurs k times in n trials p k q nk
k
k
n k nk
Pcfw_ A occurs k1 to k2 times in n trials p q
k k k
The Normal Approximation (DeMoivre-Laplace Theo

Chapter 3: Repeated Trials
Combined Experiments
Example
We are given two independent experiments.
First: rolling a fair die,
S1 cfw_ f1 , f 2 , . , f 6 , P 1cfw_ f i 1/ 6
Second: tossing a fair coin,
S2 cfw_h, t, P2 cfw_h P2 cfw_t 1/ 2
Pcfw_"two" on die,

4.4 Conditional Distribution
Recall the conditional probability of A assuming M:
P AM
P A M
where P M 0.
PM
Similarly, we define the conditional distribution of the random
variable X assuming M as
P X x , M
F x M P X x M
P M
We define the conditiona

Appendix: Set Theory
Definition
A set is a collection of objects called elements.
Example
A cfw_1 , 2 ,., n , i A, i 1, 2,., n .
A is a set whose elements are 1 , 2 ,., n .
Definition
A subset B of a set A is another set whose elements are also elements
o