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Differences Between PDF and PMF


This topic is quite complicated as it would require further understanding of more than a limited knowledge of physics. In this article, we will be differentiating PDF, probability density function, versus PMF, probability mass function. Both terms are related to physics or calculus, or even higher math; and for those taking up courses or who may be an undergraduate of math related courses, it is to be able to properly define and put a distinction between both terms so it would be better understood.

Random variables are not quite fully understandable, but, in a sense, when you talk about using the formulas that derive the PMF or PDF of your final solution, it is all about differentiating the discrete and continuous random variables that make the distinction.

The term probability mass function, PMF, is about how the function in the discrete setting would be related to the function when talking about continuous setting, in terms of mass and density. Another definition would be that for the PMF, it is a function that would give an outcome of a probability of a discrete random variable that is exactly equal to a certain value. Say for example, how many heads in 10 tosses of a coin.

Now, let’s talk about the probability density function, PDF. It is defined only for continuous random variables. What is more important to know is that the values that are given are a range of possible values that gives the probability of the random variable that falls within that range. Say, for example, what is the weight of females in California from the ages of eighteen to twenty-five.

With that as a foundation, it is easier to realize when to use the PDF formula and when you should be using the PMF formula.


In summary, the PMF is used when the solution that you need to come up with would range within numbers of discrete random variables. PDF, on the other hand, is used when you need to come up with a range of continuous random variables.
PMF uses discrete random variables.

PDF uses continuous random variables.

Based on studies, PDF is the derivative of CDF, which is the cumulative distribution function. CDF is used to determine the probability wherein a continuous random variable would occur within any measurable subset of a certain range. Here is an example:

We shall compute for the probability of a score between 90 and 110.
P (90 < X < 110)
= P (X < 110) – P (X < 90)
= 0.84 -0.16
= 0.68
= 68%

In a nutshell, the difference is more on the association with continuous rather than discrete random variables. Both terms have been used often in this article. So it would be best to include that these terms really mean.

Discrete random variable = are usually count numbers. It takes only a countable number of distinct value, like, 0,1,2,3,4,5,6,7,8,9, and so on. Other examples of discrete random variables could be:
The number of children in the family.
The number of people watching the Friday late night matinee show.
The number of patients on New Year’s Eve.

Suffice to say, if you talk about probability distribution of a discrete random variable, it would be a list of probabilities that would be associated to the possible values.

Continuous random variable = is a random variable that actually covers infinite values. Alternately, that is why the term continuous is applied to the random variable because it can assume all of the possible values within the given range of the probability. Examples of continuous random variables could be:

The temperature in Florida for the month of December.
The amount of rainfall in Minnesota.
The computer time in seconds to process a certain program.

Hopefully, with these definition of terms included in this article, it will not only be easier for anyone reading this article to understand the differences between Probability Density Function versus the Probability Mass Function.

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1 Comment

  1. Thanks for the clarification

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