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.
![probability mass function property 1](https://datasciencelk.com/wp-content/uploads/2020/03/PMF1.png)
![probability mass function property 2](https://datasciencelk.com/wp-content/uploads/2020/03/PMF2.png)
![probability mass function property 3](https://datasciencelk.com/wp-content/uploads/2020/03/PMF3.png)
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;
![table for pmf](https://datasciencelk.com/wp-content/uploads/2020/03/table.png)
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.
![Probability Mass function graph](https://datasciencelk.com/wp-content/uploads/2020/03/pmf-graph.png)
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.