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Discrete probability distributions
A random variable is said to be discrete when it can only take a finite number
of values. For example, the number of rainy days in a given location can be
considered a discrete random variable because we count them as integer
numbers only. Let X represent a discrete random variable, its probability mass
function (pmf) is represented by f(x) = P[X=x], i.e., the probability that the
random variable X takes the value x.
The mass distribution function must satisfy the conditions that
f(x) >0, for all x,
and
0 . 1 ) ( =
∑
x all
x f
A cumulative distribution function (cdf) is defined as
∑
≤
= ≤ =
x k
k f x X P x F ) ( ] [ ) (
Next, we will define a number of functions to calculate discrete probability
distributions. We suggest that you create a sub-directory, say,
HOME\STATS\DFUN (Discrete FUNctions) where we will define the
probability mass function and cumulative distribution function for the binomial
and Poisson distributions.
Binomial distribution
The probability mass function of the binomial distribution is given by
n x p p
x
n
x p n f
x n x
,..., 2 , 1 , 0 , ) 1 ( ) , , ( = − ⋅ ⋅




=
−
where (
n
x
) = C(n,x) is the combination of n elements taken x at a time. The
values n and p are the parameters of the distribution. The value n represents
the number of repetitions of an experiment or observation that can have one
of two outcomes, e.g., success and failure. If the random variable X
represents the number of successes in the n repetitions, then p represents the