9
CONVERGENCE IN PROBABILITY
9
112
Convergence in probability
One of the goals of probability theory is to extricate a useful deterministic quantity out of a
random situation. This is typically possible when a large number of random eects cancel each
othe
8
MORE ON EXPECTATION AND LIMIT THEOREMS
8
87
More on Expectation and Limit Theorems
Given a pair of random variables (X, Y ) with joint density f and another function g of two
variables,
Eg (X, Y ) =
g (x, y )f (x, y ) dxdy ;
if instead (X, Y ) is a disc
7
JOINT DISTRIBUTIONS AND INDEPENDENCE
7
70
Joint Distributions and Independence
Discrete Case
Assume that you have a pair (X, Y ) of discrete random variables X and Y . Their joint probability
mass function is given by
p(x, y ) = P (X = x, Y = y )
so tha
6
CONTINUOUS RANDOM VARIABLES
6
58
Continuous Random Variables
A random variable X is continuous if there exists a nonnegative function f so that, for every
interval B ,
P (X B ) =
f (x) dx,
B
The function f = fX is called the density of X .
We will assum
5
5
DISCRETE RANDOM VARIABLES
45
Discrete Random Variables
A random variable is a number whose value depends upon the outcome of a random experiment.
Mathematically, a random variable X is a real-valued function on , the space of outcomes:
X : R.
Sometime
4
4
CONDITIONAL PROBABILITY AND INDEPENDENCE
24
Conditional Probability and Independence
Example 4.1. Assume that you have a bag with 11 cubes, 7 of which have a fuzzy surface and
4 are smooth. Out of the 7 fuzzy ones, 3 are red and 4 are blue; out of 4 s
3
3
AXIOMS OF PROBABILITY
11
Axioms of Probability
The question here is: how can we mathematically dene a random experiment? What we have
are outcomes (which tell you exactly what happens), events (sets containing certain outcomes),
and probability (which
2
2
COMBINATORICS
3
Combinatorics
Example 2.1. Toss three fair coins. What is the probability of exactly one Heads (H)?
There are 8 equally likely outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Out
of these, 3 have exactly one H. That is, E = cfw_HTT,
1
1
INTRODUCTION
1
Introduction
The theory of probability has always been associated with gambling and many most accessible
examples still come from that activity. You should be familiar with the basic tools of the
gambling trade: a coin, a (six-sided) di
10
MOMENT GENERATING FUNCTIONS
10
120
Moment generating functions
If X is a random variable, then its moment generating function is
(t) = X (t) = E (etX ) =
tx
x e P (X = x)
tx
e fX (x) dx
in the discrete case,
in the continuous case.
Example 10.1. Assu
