CDF vs. PDF: What is the Difference?
Cumulative Distribution Function or CDF and the Probability Density Function or PDF are important in statistics when dealing with continuous random variables. While both functions provide insights into probabilities, they have different purposes and give different perspectives on the distribution of data.
In this article we will discuss about the difference between Cumulative Distribution Function and the Probability Density Function in detail.
Table of Content
- What is a PDF?
- What is a CDF?
- Difference Between CDF and PDF
- Relation Between PDF and CDF
What is a PDF?
PDF stands for Probability Density Function. It is an important concept in statistics for understanding probabilities related to continuous random variables. It is a smooth curve that shows how likely different outcomes are within a range of values.
For example, consider the temperature in a city on a given day. The PDF could show the likelihood of temperatures falling within certain ranges, like between 70°F and 80°F.
PDF does not give the probability of specific values, but rather the probability of the variable falling within a small interval around a particular value. The area under the PDF curve for a range of values represents the probability of the variable falling within that range. To find the probability of a single value, it requires to calculate the integral of the PDF at that point, which means finding the area under the curve at that specific value.
What is a CDF?
CDF stands for cumulative distribution function. The CDF complements the Probability Density Function and provides a cumulative view of the probabilities linked to a random variable. Unlike the smooth curve of the PDF, the CDF appears as a step function, jumping at specific values. It shows the probability that a random variable will be less than or equal to a given value.
Starting from 0 for negative values, the CDF gradually increases to 1 as the value of the random variable increases. For discrete random variables, the CDF rises in steps, corresponding to the probabilities of each possible outcome. With continuous random variables, it increases smoothly and reflects the combined probabilities across different intervals.
Difference Between CDF and PDF
The difference between CDF and PDF can be understood from the table given below.
Cumulative Distribution Function | Probability Density Function |
|---|---|
It gives the probability that a random variable is less than or equal to a specific value. | It describes the likelihood of a random variable falling within a small interval around a particular value. |
It represents cumulative probabilities, showing how probabilities accumulate as you move along the variable’s range. | It represents probabilities as a smooth curve, indicating how likely different outcomes are within a range of values. |
It provides the probability for specific values or ranges of values. | It does not give probabilities for specific values but indicates the probability density around each value. |
It is always non-decreasing, which means, as the variable’s value increases, the probability also increases or remains the same. | It can take any value within its range, but the total area under the curve must sum up to 1. |
It is used to calculate probabilities for specific events or intervals. | It is used to understand the overall distribution of probabilities for a continuous random variable. |
Relation Between PDF and CDF
The relationship between CDF and PDF is described below:
- The PDF describes the relative likelihood of a continuous random variable taking on a particular value.
- The CDF, on the other hand, gives the probability that a continuous random variable is less than or equal to a specified value.
Mathematically, the relationship between the PDF and CDF is as follows:
[Tex]F(x) = \int_{-\infty}^{x} f(t) dt[/Tex]
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FAQs on Difference between CDF and PDF
What is the difference between the CDF and PDF?
The key difference between the CDF and PDF depends on their representation and purpose.
- The CDF provides the probability that a random variable is less than or equal to a specific value, That represents cumulative probabilities along the variable’s range.
- On the other hand, the PDF describes the likelihood of a random variable falling within a small interval around a particular value, presenting probabilities as a smooth curve without giving specific probabilities for individual values.
What is the relationship between CDF and PDF?
The Cumulative Distribution Function (CDF) of a continuous random variable gives the probability that the variable takes a value less than or equal to a specific point. The Probability Density Function (PDF) is the derivative of the CDF and represents the density of the probability at each point.
What does CDF stand for?
CDF stands for Cumulative Distribution Function.
What are the properties of CDF?
The properties of CDF are:
- The CDF is a non-decreasing function.
- It ranges from 0 to 1.
- As the variable approaches negative infinity, the CDF approaches 0.
- As the variable approaches positive infinity, the CDF approaches 1.
Can CDF be greater than 1?
No, the CDF cannot be greater than 1. It represents a probability, which ranges between 0 and 1.
Can a PDF be 0?
Yes, a PDF can be 0 at specific points or over intervals where the probability density is zero. For instance, in a distribution where certain values cannot occur, the PDF at those values will be 0.