Probability Mass function is the function that maps the values of a discrete random variable to their probabilities.
Suppose x1, x2, ....... are possible values of a discrete random variable X. Then p(xi) is called the probability mass function of the random variable X if,

  1. p(x) ≥ 0 for all i = 1,2,3.....
  2. $\sum_{i}p(x_{i})$ = 1.

In the simple example of the random variable X assuming the number of heads in a single toss of a coin,
X = {0, 1}
p(x) is the function that gives the probabilities of X = 0 and X = 1 in a single toss.

p(0) = p(1) = $\frac{1}{2}$ and p(0) + p(1) = $\frac{1}{2}$ + $\frac{1}{2}$ = 1

The formulas used for computing the probabilities of various discrete random variables are given below.

Binomial probability formula
In a Binomial distribution of n independent trials, the probability of exactly X successes is given by
P(X : n, p) = C(n, X). pX .qn-X where p is the probability of success in a single trial and q = 1-p.

Poisson probability formula
The probability of X occurrences in an interval of time, area, volume etc is
P(X : λ) = $\frac{e^{\lambda }\lambda ^{X}}{X!}$ where λ is the mean number of occurrence per unit.

Geometric probability formula
P(X : p) = qX . p where 0 < p < 1 and q = 1 - p