Difference Between PDF and PMF

In order to understand the difference between PDF and PMF, it is important to understand what Random variables are. A random variable is a variable whose value is not known to the task; in other words, the value depends on the result of the experiment.


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For instance, while flipping a coin, the value i.e. heads or tails depends upon the outcome.

Key Takeaways

  1. PDF (Probability Density Function) is a statistical function used to describe the probabilities of continuous random variables within a given range.
  2. PMF (Probability Mass Function) is a statistical function that describes the probabilities of discrete random variables, assigning a probability to each possible outcome.
  3. PDF and PMF represent the probability distributions of random variables, but they differ in their application, with PDF used for continuous variables and PMF for discrete variables.


The difference between PDF and PMF is in terms of random variables. PDF is relevant for continuous random variables while PMF is relevant for discrete random variable.

PDF vs PMF 1

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Both the terms, PDF and PMF are related to physics, statistics, calculus, or higher math. PDF (Probability Density Function) is the likelihood of the random variable in the range of discrete value.

On the other hand, PMF (Probability Mass Function) is the likelihood of the random variable in the range of continuous values.


Comparison Table

Parameter of ComparisonPDFPMF
Full formProbability Density FunctionProbability Mass Function
UsePDF is used when there is a need to find a solution in a range of continuous random variables.PMF is used when there is a need to find a solution in a range of discrete random variables.
Random VariablesPDF uses continuous random variables.PMF uses discrete random variables.
FormulaF(x)= P(a < x 0p(x)= P(X=x)
SolutionThe solution falls in the radius range of continuous random variablesThe Solutions falls in the radius between numbers of discrete random variables


What is PDF?

The Probability Density Function (PDF) depicts probability functions in terms of continuous random variable values presenting in between a clear range of values.

It is also known as a probability distribution function or a probability function. It is denoted by f(x). 

The PDF is essentially a variable density over a given range. It is positive/non-negative at any given point in the graph and the whole of PDF is always equal to one.

In a case where the probability of X on some given value x (continuous random variable) is always 0. In such a case P(X = x) does not work.

In such a situation, we need to calculate the probability of X resting in an interval (a, b) along with for P(a< X< b) which can take place using a PDF.

The Probability distribution function formula is defined as, F(x)= P(a < x < b)= ∫ba f(x)dx>0

Some instances where Probability distribution function can work are:

  1. Temperature, rainfall and overall weather
  2. Time computer takes to process input and give output

And many more.

Various applications of the probability density function (PDF) are:

  1. The PDF is used in shaping the data of atmospheric NOx temporal concentration yearly.
  2. It is treated to shaped the diesel engine combustion
  3. It is used to work on the probabilities attached with random variables in statistics.
pdf 1

What is PMF?

The Probability Mass function depends on the values of any real number. It does not go to the value of X which equals to zero and in case of x, the value of PMF is positive.

The PMF plays an important role in defining a discrete probability distribution and produces distinct outcomes. The formula of PMF is p(x)= P(X=x) i.e the probability of (x)= the probability (X=one specific x)

As it gives distinct values, PMF is very useful in computer programming and shaping of statistics.

In simpler terms, probability mass function or PMS is a function that is associated with discrete events i.e. probabilities related with those events occurring.

The word “mass“ explains the probabilities that are focused on discrete events.

Some of the applications of the probability mass function (PMF) are:

  1. Probability mass function (PMF) has a main role in statistics as it helps in defining the probabilities for discrete random variables.
  2. PMF is used to find the mean and variance of the distinct grouping.
  3. PMF is used in binomial and Poisson distribution where discrete values are used.

Some instances where Probability mass function can work are:

  1. Number of students in a class
  2. Numbers on a dice
  3. Sides of a coin
  4. And many more.

Main Differences Between PDF and PMF 

  1. The full form of PDF is Probability Density Function whereas the full form of PMF is Probability Mass Function
  2. PMF is used when there is a need to find a solution in a range of discrete random variables whereas PDF is used when there is a need to find a solution in a range of continuous random variables.
  3. PDF uses continuous random variables whereas PMF uses discrete random variables.
  4. Pdf formula is F(x)= P(a < x < b)= ∫ba f(x)dx>0 whereas pmf formula is p(x)= P(X=x)
  5. The solutions of PDF falls in the radius of continuous random variables whereas the solutions of PMF falls in the radius between numbers of discrete random variables

  1. https://amstat.tandfonline.com/doi/abs/10.1080/10485250701733747
  2. https://www.mitpressjournals.org/doi/abs/10.1162/0899766053723078
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