PDF vs PMF: Difference and Comparison

To understand the difference between PDF and PMF, it is essential 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.

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.

PDF vs PMF

PDF, also known as the probability density function, is a mathematical function that is used when there is a solution to be found within a range of continuous random variables. PMF, also known as probability mass function is a function that used discrete random variables to find a solution.

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 values.

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

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Comparison Table

Parameter of Comparison	PDF	PMF
Full form	Probability Density Function	Probability Mass Function
Use	PDF is used when there is a need to find a solution in a range of continuous random variables.	PMF is used when finding a solution in a range of discrete random variables is needed.
Random Variables	PDF uses continuous random variables.	PMF uses discrete random variables.
Formula	F(x)= P(a < x 0	p(x)= P(X=x)
Solution	The solution falls in the radius range of continuous random variables	The Solutions fall in the radius between numbers of discrete random variables

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What is PDF?

The Probability Density Function (PDF) depicts probability functions in terms of continuous random variable values between a precise 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 full PDF always equals one.

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

In such a situation, we need to calculate the probability of X resting in an interval (a, b) along with 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)= ∫^b_a f(x)dx>0

Some instances where the Probability distribution function can work are:

1. Temperature, rainfall and overall weather
2. Time the 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 shape the diesel engine combustion.
3. It works on the probabilities attached to random variables in statistics.

 

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 zero; in the 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 the shaping of statistics.

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

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

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

1. The probability mass function (PMF) is central in statistics as it helps define 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 distributions where discrete values are used.

Some instances where the 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.

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Main Differences Between PDF and PMF 

1. The complete 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)= ∫^b_a f(x)dx>0 whereas pmf formula is p(x)= P(X=x)
5. The solutions of PDF fall in the radius of continuous random variables, whereas the solutions of PMF fall in the radius between numbers of discrete random variables

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References

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

Last Updated : 11 June, 2023