RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
1. DISCRETE RANDOM VARIABLES
1.1. Definition of a Discrete Random Variable. A random variable X is said to be discrete if it can
assume only a finite or countable infinite number of distinct values. A discrete random variable
can be defined on both a countable or uncountable sample space.
1.2. Probability for a discrete random variable. The probability that X takes on the value x, P(X=x),
is defined as the sum of the probabilities of all sample points in Ω that are assigned the value x. We
may denote P(X=x) by p(x) or p
X
(x). The expression p
X
(x) is a function that assigns probabilities
to each possible value x; thus it is often called the probability function for the random variable X.
1.3. Probability distribution for a discrete random variable. The probability distribution for a
discrete random variable X can be represented by a formula, a table, or a graph, which provides
p
X
(x) = P(X=x) for all x. The probability distribution for a discrete random variable assigns nonzero
probabilities to only a countable number of distinct x values. Any value x not explicitly assigned a
positive probability is understood to be such that P(X=x) = 0.
The function p
X
(x)= P(X=x) for each x within the range of X is called the probability distribution
of X. It is often called the probability mass function for the discrete random variable X.
1.4. Properties of the probability distribution for a discrete random variable. A function can
serve as the probability distribution for a discrete random variable X if and only if it s values,
p
X
(x), satisfy the conditions:
a: p
X
(x) ≥ 0 for each value within its domain
b:
P
x
p
X
(x)=1, where the summation extends over all the values within its domain
1.5. Examples of probability mass functions.
1.5.1. Example 1. Find a formula for the probability distribution of the total number of heads ob-
tained in four tosses of a balanced coin.
The sample space, probabilities and the value of the random variable are given in table 1.
From the table we can determine the probabilities as
P (X =0) =
1
16
,P(X =1) =
4
16
,P(X =2) =
6
16
,P(X =3) =
4
16
,P(X =4) =
1
16
(1)
Notice that the denominators of the five fractions are the same and the numerators of the five
fractions are 1, 4, 6, 4, 1. The numbers in the numerators is a set of binomial coefficients.
1
16
=
4
0
1
16
,
4
16
=
4
1
1
16
,
6
16
=
4
2
1
16
,
4
16
=
4
3
1
16
,
1
16
=
4
4
1
16
We can then write the probability mass function as
Date: November 1, 2005.
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