PDF vs. CDF: Difference Between PDF and CDF

PDF vs. CDF: Difference Between PDF and CDF

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Feb 9, 2024 12:33 IST

Pdf describes the probability distribution of a continuous random variable, while cdf describes the probability distribution of both discrete and continuous random variables. In this article, we will learn the difference between pdf and cdf.

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The probability distribution of a random variable is a list of all possible outcomes with corresponding probability values, and different functions are used to describe this probability distribution. Probability Density Function and Cumulative Distribution Function are among such functions.

The probability Density Function describes the probability distribution of continuous random variables. Cumulative Distribution Function is a probability distribution that deals with continuous and discrete random variables.

In this article, we will learn the difference between PDF and PDF in probability.

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Before starting the article, let’s discuss random variables.

A Random Variable is a numerical description of the outcome of a random experiment. Generally, it is represented by X.

It is of two types: 

Discrete Random Variable: It has a countable number of values between two values.

Example: Number of Heads 

Continuous Random Variable: It has an infinite number of values between two values.

Example: Distance Travelled

Now, let’s start with the probability density function vs cumulative distribution function based on the different parameters.

Table of Content

CDF vs PDF: Difference Between PDF and CDF

Parameter PDF CDF
Definition PDF is the probability that a random variable (let X), will take a value exactly equal to the random variable (X). CDF is the probability that a random variable (let X) will take a value less than or equal to the random variable (X).
Variable Applicable only for Continuous Random Variable. Applicable for both continuous and discrete random variables.
Value The value lies between 0 and 1. The value of CDF is always non-negative.
Formula P (a <= X <= b) = F(b) – F(a) FX(X) = P (X<= X)
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What is the Probability Density Function

PDF (or Probability Density Function) is a function that describes the probability of a continuous random variable. The PDF function’s curve looks like a bell-shaped curve, and it takes the value between the given intervals. 

  • It tells the probability that a random variable takes on a certain value. 

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Formulas

The Probability density function of a continuous random variable X is a function fX: R -> [0, inf) such that:

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Where R is a real number.

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Properties

  • The value of PDF is always greater than or equal to 1.
  • The area under the PDF curve is always equal to 1.
  • The median divides the PDF curve into two equal halves.
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Example

If

f(x) = x, if 0 < x < 1

f(x) = 2 – x, if 1 < x < 2 and

f(x) = 0, otherwise

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Then, f(x) is a probability density function centred at x = 1.

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What is the Cumulative Distribution Function?

CDF or Cumulative Distribution Function is a probability that a random variable takes on a value less than or equal to X. The CDF can be defined. for both the random variables: continuous and discrete random variables.

Formula

The CDF (Cumulative Distribution Function) of a random variable X is defined as

FX(X) = P (X <= X),

for all x in R.

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Properties

  • CDF is a non-decreasing.
  • The minimum of CDF is 0 when x = – infinity: Fx(- inf) = 0.
  • The maximum of CDF is 1 when x = infinity: Fx(inf) = 1
  • P [a <= x <= b] = FX(b) – FX(a)

Example

Consider the pdf function given in the above example and find the CDF for ½ <= x <= 3/2.

Answer:

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Relation Between PDF and CDF

  • CDF can be found using PDF by simply integrating PDF.
  • PDF can be obtained using CDF by differentiating CDF.

What are the key differences between PDF and CDF?

  1. PDF is the probability that a random variable will take a value exactly equal to the random variable. In contrast, CDF is the probability that a random variable will take a value less than or equal to the random variable.
  2. PDF applies only for continuous random variables, while CDF applies for continuous and discrete random variables.
  3. The value of CDF is always non-negative, whereas the value of PDF lies between 0 and 1.
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Conclusion

In this article, we have discussed the probability density function (PDF) and cumulative density function (CDF), their properties and the difference between CDF and PDF with the help of examples.

PDF (or Probability Density Function) is a function that describes the probability of a continuous random variable. The PDF function’s curve looks like a bell-shaped curve, taking the value between the given intervals. In contrast, CDF or Cumulative Distribution Function is a probability that a random variable takes on a value less than or equal to X. The CDF can be defined. for both the random variables: continuous and discrete random variables.

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FAQs on Difference Between

What is PDF?

PDF (or Probability Density Function) is a function that describes the probability of a continuous random variable. The PDF function's curve looks like a bell-shaped curve, and it takes the value between the given intervals.

What is CDF?

CDF (or Cumulative Distribution Function) is a probability that a random variable takes on a value less than or equal to X. The CDF can be defined. for both the random variables: continuous and discrete random variables.

What is the difference between PDF and CDF?

A PDF, or Probability Density Function, shows the likelihood of a continuous random variable assuming specific values. In contrast, a CDF, or Cumulative Distribution Function, represents the probability that the variable is less than or equal to a particular value. The PDF is used more for continuous variables, while the CDF can be applied to both continuous and discrete variables.

When to use PDF and CDF?

PDF is mainly used in statistical analysis and modeling to describe the distribution of data, whereas CDF are commonly used in statistical analysis and modeling to calculate probabilities and percentiles.

Can a PDF convert to a CDF?

Yes, PDF can be converted to a CDF by integrating the PDF over its entire domain. The resulting function will be the CDF of the random variable and vice versa.

How do PDF and CDF function for discrete random variables?

For discrete random variables, the PDF (in this case, more accurately referred to as a Probability Mass Function or PMF) outlines the probabilities for each specific value. The CDF accumulates these probabilities, showing the cumulative probability for values up to and including a specific point.

Can you calculate probabilities using both PDF and CDF?

Yes, both can be used to calculate probabilities, but in different ways. For continuous random variables, the PDF can find cumulative probabilities over a range, whereas the CDF gives the cumulative probability up to a specific value.

About the Author
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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio