# Probability Mass Function

## Introduction to PMF

Probability Mass Function, PMF of the Random Variable X says how the total probability of 1 is distributed or allocated to among the various possible X values.

## Definition of Probability Mass Function

The Probability Mass Function, P(X = x), f(x) of a discrete random variable X is a function that satisfies the following properties.

## Descriptive Example

Consider a Mobile Phone manufacturing factory. Suppose they have prepared 5 boxes of Mobile Phones to be delivered to 5 different customers.

The number of defective mobile phones in each box are as follows;

One of these boxes will be selected to send to a customer.

Let X be the number of defectives in the selected box.

3 possible values of X are 0, 1 and 2.

These are 5 equally likely simple events.

#### Probability Mass Function of X is;

p(0) = P(X=0) = P(box 1 or 2 sent) = 2/5 = 0.4

p(1) = P(X=1) = P(box 3 or 5 sent) = 2/5 = 0.4

p(2) = P(X=2) = P(box 4 sent) = 1/5 = 0.2

### Interpretation of Probability Mass Function

A probability of 0.4 is distributed to values 0 and 1. And a probability of 0.2 is distributed to value 2.

Values of X along with their probabilities define the PMF.

### Graphical Representation of PMF

We can plot the values of p(x) against each value of x.

Therefore it is clear that PMF can be interpreted both Numerically and Visually.

## Why it is called Probability Mass Function?

Name PMF is suggested by a model used in Physics for a system of Point Masses. In this model, masses are distributed at various locations X along a one-dimensional axis.

Our PMF describes how the total probability mass of 1 is distributed at various points along the axis of possible values of the Random Variable X (where and how much mass at each x).

Concept of Probability Mass Function will be very useful when you are studying further probability, expected values and so on.