A PRIMER FOR
RANDOM VARIABLES AND
PROBABILITY DISTRIBUTIONS
by
Dr. Harvey A. Singer
1 Introduction
A random variable is a variable that takes on numerical values determined by the
outcome of a random experiment.
A random variable is discrete if it can take on no
more than a countable number of
values.
Discrete random variables generally result from a counting process.
A random
variable is continuous if it can take any value in an interval.
Continuous random
variables generally result from a measuring process.
The two obvious questions regarding random variables are:
(1) What is typical or expected
of the random variable?
(2) What are the probabilities that particular values or ranges of values will occur?
The concept of expected value addresses questions of the first type, and the concept of
probability distributions addresses questions of the second type.
A probability distribution for a random variable, whether discrete or continuous, may
generally be thought of as a theoretical listing of outcomes and their probabilities, which
often may be obtained from a mathematical model representing the phenomenon of
interest.
A model is idealized representation of some underlying phenomenon.
In particular, a
mathematical model is a mathematical expression representing some underlying
phenomenon.
For discrete random variables, this mathematical expression is known as a
probability distribution function
.
A probability distribution for a discrete random
variable is simply a listing of the probabilities of all possible outcomes, according to the
values of the random variable.
For continuous random variables, the mathematical model
representing some underlying phenomenon is known as a
probability density function
.
When such a mathematical expression is available, the probability that various values of
the random variable occur within specified ranges or intervals may be calculated.
© 1999 by Ha
rvey A. Singer
1
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View Full Document(However the exact probability of a particular value is zero.)
2 Probability
Distributions
for
Discrete
Random
Variables
2.1 Definition
A probability distribution (or probability function or probability distribution function) for
a discrete random variable is a mutually exclusive listing of all possible outcomes for that
random variable such that a particular probability of occurrence is associated with each
occurrence.
In particular, the probability distribution of a discrete random variable
X
expresses the probability that
X
takes on the particular value
x
, as a function of
x
.
That is,
P
X
(
x
) =
P
(
X
=
x
)
2.2 Properties of Probability Functions of Discrete Random Variables
1.
P
X
(
x
) ³ 0 for any value
x
of
X
.
2. The individual probabilities sum to 1, that is,
P
X
x
(
29
=
1
x
∑
where the summation is over all possible values
x
of
X
.
2.3 Cumulative Probability Functions
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 Fall '08
 SINGER
 Normal Distribution, Poisson Distribution, Probability theory, probability density function, Dr. Harvey A. Singer

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