STAT 421 Lecture Notes
3.3
40
Cumulative Distribution Functions
Denition The cumulative distribution function (c.d.f.) of a random variable X is the
function
F (x) = Pr(X x) x R.
The c.d.f. exists and is dened in the same manner for all random variables (
STAT 421 Lecture Notes
3.4
45
Bivariate Distribution
Denition 3.4.1 Suppose that X and Y are random variables. The joint distribution, or bivariate distribution of X and Y is the collection of all probabilities of the form Pr[(X, Y ) C ]
for all sets C R2
STAT 421 Lecture Notes
3.5
52
Marginal Distributions
Denition 3.5.1 Suppose that X and Y have a joint distribution. The c.d.f. of X derived
by integrating (or summing) over the support of Y is called the marginal c.d.f. of X . The
p.f. or p.d.f. associate
STAT 421 Lecture Notes
3.2
36
Continuous Distributions
In the previous section, the emphasis was upon discrete random variables. Attention shifts
now to continuous random variables. Earlier it was said that a continuous random variable
X diered from discr
STAT 421 Lecture Notes
1
Introduction
1
Introduction to Probability
Probability is used to describe the possible results of an action with an unknown outcome.
Specically, the aim is to assign a numerical value to every possible outcome that reects the
lik
STAT 421 Lecture Notes
3
32
Chapter 3: Random Variables
A random variable is a real-valued function dened on a sample space S ; hence, if X is a
random variable, then X : S R.
For example, suppose that an experiment consists of tossing four fair coins. Th
STAT 421 Lecture Notes
2
18
Chapter 2: Conditional Probability
Consider a sample space S and two events A and B . For example, suppose that the equally
likely sample space is S = cfw_0, 1, 2, . . . , 99 and A is the event that the outcome is at least
50 a
STAT 421 Lecture Notes
3.8
76
Functions of a Random Variable
This section introduces a new and important topic: determining the distribution of a function of a random variable. We suppose that there is a random variable X and a function
r dened on the sup
STAT 421 Lecture Notes
3.6
58
Conditional Distributions
Denition 3.6.1. Suppose that X and Y have a discrete joint distribution with joint p.f. f
and let f2 denote the marginal p.f. of Y . For each y such that f2 (y ) > 0, the conditional p.f.
of X given
STAT 421 Lecture Notes
4.3
102
Variance
Denition Let X be a random variable with nite mean = E (X ). The variance of X is
dened to be
2 = Var(X ) = E [(X )2 ].
If X has innite mean, or the mean does not exist, then we say that the variance of X does
not
STAT 421 Lecture Notes
4.4
105
Moments
Let X be a random variable and k denote a positive integer. The k th moment of X is dened
to be E (X k ). The k th moment need not exist, though the k th and lower moments exist if
E (|X |k ) exists.
Theorem 4.4.1. I
STAT 421 Lecture Notes
3.9
82
Functions of Two or More Random Variables
This section discusses methods of determining the distribution of a function of several random
variables. The subject is important in statistics because a statistic is a function of (
STAT 421 Lecture Notes
4.2
98
Properties of Expectation
Theorem 4.2.1. Suppose that X is a random variable for which E (X ) exists. Let Y = a + bX
where a, b R. Then,
E (Y ) = a + bE (X ).
Let S denote the support of X . The proof for the discrete case pr
STAT 421 Lecture Notes
3.7
64
Multivariate Distributions
This section extends denitions and concepts from the previous sections on bivariate distributions to distributions of more than two random variables. There are very few new concepts.
Denition 3.7.1.
STAT 421 Lecture Notes
4.1
93
The Expectation of a Random Variable
This chapter begins the discussion of properties of random variables. The focus of this chapter is on expectations of random variables.
Example Suppose that X has a discrete uniform distri