Vikram Singh
Assistant Manager - Content
Updated on Apr 24, 2023 17:01 IST

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

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)! ] * px * (1-p)n-x

Poisson Distribution

P(X = x) = e * λx / (x!),

where,

x = 0, 1, 2, 3, …

lambda is any positive number.

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!!