# PDF, CDF and PMF – Probability Distribution Functions

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In probability and statistics, several terms are used to describe the various functions that are used to model probability distributions. These include:

## Probability density function (PDF):

The PDF is a function that describes the probability of a continuous random variable taking on a certain value. It is a mathematical function that describes the probability that a random variable will fall within a certain range of values.

## Cumulative distribution function (CDF):

The CDF is a function that describes the probability that a random variable (continuous or discrete) will take on a value less than or equal to a certain value. It is a mathematical function that describes the probability that a random variable will fall within a certain range of values, up to and including a specific value.

## Probability mass function (PMF):

The PMF is a function that describes the probability of a discrete random variable taking on a certain value. It is a mathematical function that describes the probability that a random variable will take on a specific value rather than falling within a range of values.

## Summary:

In summary, PDFs are used to describe the probability of a continuous random variable taking on a certain value, CDFs are used to describe the probability that a random variable (continuous or discrete) will take on a value less than or equal to a certain value, and PMFs are used to describe the probability of a discrete random variable taking on a certain value.

Feature Probability Density Function (PDF) Cumulative Distribution Function (CDF) Probability Mass Function (PMF)
Function of Describes the likelihood of a continuous random variable taking on a specific value or falling within a specific interval Describes the probability that a random variable (continuous or discrete) will take on a value less than or equal to a certain value Describes the probability of a discrete random variable taking on a specific value
Types of Variables Continuous random variables Continuous and discrete random variables Discrete random variables
Probability The value of the PDF at a specific point indicates the probability density, not the probability itself The value of the CDF at a specific point indicates the probability that the random variable will take on a value less than or equal to that point The value of the PMF at a specific point indicates the probability of the random variable taking on that value
Sum or Integral The integral of the PDF over the entire range of the random variable equals 1 The limit of the CDF as the random variable approaches positive infinity is 1 The sum of the PMF over all possible values of the random variable equals 1
Visualization The PDF can be visualized as a curve, with the area under the curve representing the probability The CDF can be visualized as a curve that starts at 0 and increases monotonically to 1 The PMF can be visualized as a bar chart, with each bar representing the probability of a specific value

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