What is Joint Probability Density Function or Joint PDF?
Joint PDF is simply the PDF of two or more random variables.The joint probability density function of any two random variables X and Y can be defined as the partial derivative of the joint cumulative distribution function, with respect to dummy variables x and y.
Mathematically-
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Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof
Joint PDF Formula
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| Joint PDF Formula |
Now we will discuss the properties of joint probability density function (joint PDF)
Properties of Joint Probability Density Function (Joint PDF)
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| Joint PDF Property |
Since joint PDF is a derivative of joint cumulative distribution function (Joint CDF), which is also a non negative function.
Therefore joint PDF is always positive.
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Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF
Property 2- The joint PDF is continuous everywhere as the joint CDF is continuous and we know that it is the derivative of joint CDF.
Property 3- The total volume under the surface of joint PDF is equal to Unity.
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| Joint PDF Property |
Now we will discuss an important property of two statistically independent random variables X and Y.
Statistically Independent Random Variables X and Y
For two statistically independent random variables X and Y- |
| Statistically independent random variables X and Y |
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Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF
Proof-
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| Statistically Independent Random Variables X and Y |
Relation between Probability and Joint PDF
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| Relation between Probability and Joint PDF |
On extending this relation to two random variables X and Y-
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| Relation between Probability and Joint PDF for two Random Variables |
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Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation
Relationship between joint PDF and Probability for statistically independent random variables X and Y
If two random variables X and Y are statistically independent, then the joint PDF of X and Y is given as the product of two separate PDFs.
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| Relationship between joint PDF and Probability for statistically independent random variables X and Y |
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| Relationship between joint PDF and Probability for statistically independent random variables X and Y |
