All About Probability Mass Function

Discover the fundamentals of probability mass function in this comprehensive guide. Learn what a PMF is, and how to use PMFs to calculate probabilities. You’ll also explore the relationship between PMFs and other statistical concepts, such as binomial distributions, and Poisson distributions. Whether you’re a beginner or an experienced data analyst, this article has everything you need to understand and apply PMFs to real-world problems.

In the previous articles, we discussed different types of probability distribution and one of the probability distribution functions that deal with the continuous random variable, i.e., the Probability Density Function (or PDF). In continuation, we will discuss another probability distribution function, i.e., the Probability Mass function (or PMF).

The Probability Mass Function is a function that gives the probability of each possible outcome in the probability distribution of the discrete random variable. 

So without further delay, let’s start.

Table of Content

• Probability Mass Function
  • Examples
  • Formula
• PDF vs PMF
• Applications
• Limitations

What is Probability Mass Function?

PMF is a function that describes the probability of each possible outcome of a random variable, i.e., it represents the probability that a random variable takes on each possible outcome in a sample space.

Mathematically it is given as:

P (X = x) = f(x), 

where

X: Random Variable

x: The specified value of the random variable at any time (or the value that the random variable can take)

P(X = x): Probability of a random variable X at X = x

Note: The above formula (or PMF) must satisfy two conditions

1. Non-Negativity: The value of PMF must always be greater than or equal to 0.
2. Normalization: The sum of all possible values of the random variable equals 1.

Now, let’s take some examples to understand it better.

Examples of Probability Mass Function

Example-1: Coin Tosses – Consider the case of flipping a coin.

Solution:

The probability of getting the number of head = probability of getting the number of tails = 0.5.

In this case, the PMF will be:

P (X = Head) = P (X = Tail) = 0.5

Example-2: Dice Roll – Consider the case of rolling a fair six-sided coin.

Solution:

The probability of getting each possible outcome of rolling dice equals 6.

In this case, the probability mass function will be:

P (X = 1) = P (X = 2) = P (X = 3) = P (X = 4) = P (X = 5) = P (X = 6) = ⅙.

Example-3: Determine the value of constant c so that the function f(x) ( f(x) = cx³, x = 1, 2, 3) satisfies the condition of being PMF.

Solution:

From the definition of probability mass function, we have a sum of all possible values of PMF for different random variable equals 1.

i.e., Σf(x) = 1

Now, substituting the value of f(x) for x = 1, 2, 3, we get:

c(1)³ + c(2)³ + c(3)³ = 1

=> c[1 + 8 + 27] = 1 => 36c = 1

=> c = 1/36

Now, it’s time to get the formulas of probability mass function for different probability distributions.

Probability Mass Function Formula

Binomial Distribution

f(x; n, p) = [n! / x! *  (n – x)! ] * p^x * (1-p)^n-x

Poisson Distribution

P(X = x) = e^-λ * λ^x / (x!), 

where,

x = 0, 1, 2, 3, …

lambda is any positive number.

Difference Between PMF and PDF

Probability Mass Function	Probability Density Function
Defined for the discrete random variable.	Defined for the continuous random variable.
Gives the probability of a specific value.	Gives the probability of a range of values.
Represented as a series of discrete bars.	Represented as a continuous curve.
Values are non-negative and add up to 1.	Values can be greater than 1, but the area under the curve adds up to 1.

Application of PMF

Domain	Application
Risk Analysis	Used to model the probability of the events, such as accidents (man-made or natural), illness etc, and to calculate the expected payout in the case of claims.
Traffic Analysis	It helps to model different traffic volume at different times of the day, which help to optimize traffic management.
Gaming	Is used to model the probabilities of different outcomes in games of chance, such as dice and card game.
Finance	Is used to predict the stock price, bond yields or interest rate.

Limitations of PMF

• Applicable only to a discrete random variable.
• It requires that possible outcomes for the random variable be exhaustive and mutually exclusive.
• Assumes the number of possible outcomes for the random variable is finite.
  • If the number of outcomes is large, it would be difficult to interpret or visualize.

Conclusion

In this article, we have discussed probability mass function which is used to describe discrete random variables. The article also covers examples, application, and limitations of PMF. The article also discusses how probability mass function differs from the probability density function.

Hope you will like the article.

Happy Learning!!