 # How to Find Probability Mass Function

Probability comes into application in the fields of physical sciences, commerce, biological sciences, medical sciences, weather forecasting, etc. As far as the JEE exam is concerned, probability is an important topic. Consider probabilities for random variables. We can calculate the probability that a discrete random variable equals a specific value. The probability mass function (PMF) shows the distribution of a discrete random variable. It associates with any given number the probability that the random variable will be equal to that number. A discrete random variable has a finite number of outcomes. The probability of each value of a discrete random variable is lies between 0 and 1. In this article, we discuss how to find probability mass function.

## What is Probability Mass Function?

If X be a discrete random variable of a function, then the probability mass function of a random variable X is given by

Px(x) = P(X = x), ∀ x ∈ range of X.

The probability function should satisfy the condition:

1. Px(x) ≥ 0

2. ∑x ∈ range( X) Px(x) = 1. The range of x has countable number of elements.

It assigns probabilities to the possible values of the random variable. It depends on the probability measure of the sample space.

## Properties

The properties of probability mass function are given below.

1. All probabilities are greater than or equal to zero. I.e. Px(x) ≥ 0.

2. The sum of the probabilities is equal to unity (1).

3. Individual probability is found by the sum of x values in the event A. P(X∈A) = ∑x∈A f(x).

4. Any event in the distribution has probability of happening between 0 to 1.

### Example

A fair coin is tossed twice. Let X be defined as the no. of heads shown. Find the range of X and its probability mass function PX.

Solution:

The sample space S = {HH, TT, HT, TH}

The no. of heads can be 0,1,or 2.

So R(x) = {0,1,2}

The probability mass function PX(x) = P(X = x) for x = 0,1,2

PX(0) = P(X = 0) = ¼ [ i.e. probability of getting no head]

PX(1) = P(X = 1) = ¼ + ¼ = ½

PX(2) = P(X= 2) = ¼