Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

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

Joint PDF Formula

Now we will discuss the properties of joint probability density function (joint PDF)

Properties of Joint Probability Density Function (Joint PDF)

Property 1- Joint PDF is non-negative.

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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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.

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-

Statistically Independent Random Variables X and Y

Relation between Probability and Joint PDF

Relation between Probability and Joint PDF

On extending this relation to two random variables X and Y-

Relation between Probability and Joint PDF for two Random Variables

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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.

Relationship between joint PDF and Probability for statistically independent random variables X and Y

Relationship between joint PDF and Probability for statistically independent random variables 
X and Y

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Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

Cumulative Distribution Function (CDF) - Properties of CDF - CDF Definition, Basics - Continuous and Discrete CDF

Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

What is Probability Density Function (PDF)?

The derivative of Cumulative Distribution Function (CDF) w.r.t. some dummy variable is called as probability density function (PDF).
Probability density function can be defined mathematically as-

Relation between PDF and CDF (Formula of PDF)

Now we will discuss the properties of probability density function. The derivation of properties of PDF is also provided here.

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Properties of Probability Density Function (PDF) with Derivation

Property 1- Probability density function is always non-zero for all values of x.

PDF Property 1 with Proof

Property 2- The area under the PDF curve is always equal to Unity i.e. one.

PDF Property 2 With Proof

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Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

Property 3- It is possible to get Cumulative Distribution Function (CDF) by integrating PDF.

PDF Property 3 With Proof

Property 4- Probability of the event {x1< X<= x2} is given by the area under the PDF curve in 
{x1< X<= x2} range.

PDF Property 4 With Proof

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Random Variables (Discrete and Continuous Random Variables), Sample space and Random Variables Examples

Probability Density Function (PDF) - Definition, Basics and Properties of Probability Density Function (PDF) with Derivation and Proof

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Joint Probability Density Function (Joint PDF) - Properties of Joint PDF with Derivation- Relation Between Probability and Joint PDF

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Conditional Probability Density Function (Conditional PDF) - Properties of Conditional PDF with Derivation

Marginal Probability Density Function (Marginal PDF) - Marginal Densities with Derivation and Proof