CDF vs. PDF
What's the Difference?
The Cumulative Distribution Function (CDF) and Probability Density Function (PDF) are both mathematical functions used in probability theory to describe the distribution of a random variable. The CDF gives the probability that a random variable takes on a value less than or equal to a given value, while the PDF gives the probability density at a specific value. In other words, the CDF provides cumulative probabilities, while the PDF provides the likelihood of a specific outcome. Both functions are essential tools in understanding the behavior of random variables and are often used in statistical analysis and modeling.
Comparison
| Attribute | CDF | |
|---|---|---|
| Definition | Cumulative Distribution Function | Probability Density Function |
| Range | 0 to 1 | 0 to infinity |
| Interpretation | Probability that a random variable is less than or equal to a certain value | Probability density at a specific value |
| Area under curve | Always equal to 1 | Not necessarily equal to 1 |
| Used for | Calculating probabilities of random variables | Describing the likelihood of a random variable taking on a specific value |
Further Detail
Definition
Cumulative Distribution Function (CDF) and Probability Density Function (PDF) are two important concepts in probability theory and statistics. CDF is a function that gives the probability that a random variable X will be less than or equal to a certain value x. It provides a complete picture of the distribution of a random variable. On the other hand, PDF is a function that describes the likelihood of a random variable taking on a particular value. It is used to describe the probability distribution of a continuous random variable.
Representation
CDF is represented by a graph or a mathematical formula that shows the cumulative probability of a random variable. It starts at 0 and ends at 1, as the probability of any random variable being less than itself is 1. PDF, on the other hand, is represented by a curve on a graph. The area under the curve represents the probability of the random variable falling within a certain range. The curve may not touch the x-axis, as it represents probability density rather than probability itself.
Properties
CDF is a non-decreasing function, meaning that as the value of x increases, the value of the CDF also increases or remains constant. It ranges from 0 to 1, with a value of 0 at negative infinity and a value of 1 at positive infinity. PDF, on the other hand, is non-negative for all values of x and integrates to 1 over the entire range of possible values. This ensures that the total probability of all possible outcomes is equal to 1.
Relationship
CDF and PDF are closely related to each other. The PDF is the derivative of the CDF. This means that if you know the PDF of a random variable, you can find the CDF by integrating the PDF. Similarly, if you know the CDF, you can find the PDF by differentiating the CDF. This relationship allows for easy conversion between the two functions and provides a comprehensive understanding of the distribution of a random variable.
Application
CDF and PDF are used in various fields such as engineering, finance, and biology. CDF is often used to calculate probabilities in statistical analysis, such as finding the probability that a random variable falls within a certain range. PDF is used to model the distribution of continuous random variables, such as the height of individuals in a population. Both functions play a crucial role in understanding and analyzing data in a wide range of applications.
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