## What is Marginal Probability Density Function (Marginal PDF) or Marginal Densities?

When the PDFs fx(x) and fy(y) for any single random variable are obtained from the joint PDF, in that case fx(x) and fy(y) are called as marginal PDF or marginal densities.

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

Proof-

Now to find the CDF of random variable X, the value of other random variable Y, does not matter.

Similarly we can get-

Here fx(x) and fy(y) are known as marginal PDF or simply marginal densities as both of these PDFs are obtained from the joint PDF.

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

## What is Conditional Probability Density Function (Conditional PDF)?

A probability density function is known as conditional PDF, when one random variable out of two random variables has a fixed value. Here suppose we have two random variables X and Y, and X has a fixed value equal to x.
In this case in the conditional PDF of Y when X=x is given as-

In the same way, Conditional PDF of X when random variable Y takes a fixed value Y=y is given as-

## Properties of Conditional Probability Density Function (Conditional PDF)

Property 1- Conditional PDF is a non-negative function.

As conditional PDF is a ratio of two PDFs and we know that PDF is a non-negative function. Therefore the ratio of two non-negative PDFs is also non- negative function.

Property 2- Area under the conditional PDF is unity i.e. one.
This property mathematically can be expressed as-

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

Property 3- Conditional probability density function (conditional PDF) reduces to marginal density if random variables X and Y are statistically independent.

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

### Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Joint Cumulative Distribution Function is also known as Joint Distribution Function or Combined CDF
Here we will discuss the CDF for two random variables X and Y.

## Definition of Joint Distribution Function (Combined CDF)

Joint CDF : [FXY(x,y)], of two random variables X and Y is defined as the probability that the random variable 'X' is less than or equal to a specified value 'x' and the random variable 'Y' is less than or equal to a specified value 'y'.

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

This definition of joint commutative distribution function (Joint Distribution Function) can be represented mathematically as-

## Properties of Joint Cumulative Distribution Function (Combined CDF)

Property 1- The joint cumulative distribution function is a monotone non-decreasing function of both x and y.

Property 2- Combined CDF is a non-negative function.
=>   Fxy(x,y) ≥ 0
It is defined as the probability in the joint sample space of random variables and probability lies between 0 and 1.
Therefore joint cumulative distribution function (Joint CDF) also lies between 0 and 1.
=> It is a non-negative function

Property 3- Joint cumulative distribution function is always continuous everywhere in the xy-plane.

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

## 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)

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.

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. 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

Joint Cumulative Distribution Function - Joint Distribution Function - Combined CDF

Proof- Statistically Independent Random Variables X and Y 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

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. 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

## 27 Dec 2018

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

Let's first understand, what is Cumulative Distribution Function (CDF) and it's Definition-

## What is Cumulative Distribution Function (CDF)?

### Definition of CDF-

The Cumulative Distribution Function (CDF) of a random variable 'X' may be defined as the probability that the random variable 'X' takes a value 'Less than or equal to x'.
Mathematically it can be represented as- CDF Formula

Cumulative Distribution Function (CDF) may be defined for-
#Continuous random variables and
#Discrete random variables

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

## Cumulative Distribution Function (CDF) other Names

#Probability distribution function of the random variable
#Distribution function of the random variable
#Cumulative probability distribution function

Following image discusses the 3 Properties of Cumulative Distribution Function (CDF) Properties of CDF (Cumulative Distribution Function Properties)

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

## Cumulative Distribution Function (CDF) for discrete random variables

If 'X' is a discrete random variable, then it takes on values at discrete points.
Therefore CDF can be defined for this case as shown in the image below- CDF for Discrete Random Variable Explanation

So the CDF for a discrete random variable for the complete range of x can be defined as shown in the image below- CDF for Discrete Random Variable

=> Cumulative Distribution Function (CDF) of a discrete variable at any certain event is equal to the summation of the probabilities of random variable upto that certain event.
As x varies from -∞ to ∞ the graph of CDF i.e. Fx(x) resembles a staircase with upward steps having height P(X=xj) at each x=xj.
But note one thing that the graph of Fx(x) CDF remains constant between the two steps or events.

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

## 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.

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

## 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

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

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

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

Before understanding the concept of random variables, you need to know what is a sample space in probability.

## What is Sample Space in probability (S)

The range of all possible outcomes of an experiment is known as the 'Sample Space' (s).

Now let's understand the definition of Random Variables-

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

## Random Variable (Random Variable Definition)

A random variable is a function which can take on any value from the sample space and having range of some set of real numbers, is known as the random variable of the experiment.

## Types of Random Variables

Random variables can be classified as-
#Discrete Random Variables and
#Continuous Random Variables

Now we will understand the Discrete Random Variables with the help of an example-

These are the random variable which can take on only finite number of values in a finite observation interval. So we can say that to discrete random variable has distinct values that can be counted.
We Will understand this with the help of an example-

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

## Example of Discrete Random Variable

Let's take an example (experiment) of tossing 3 coins at the same time (simultaneously).
Now for this experiment the sample space is
S= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Here let's suppose that the number of tails is the random variable X
So X= {0    1    1   1   2   2   2  3}
= {x1  x2  x3 x4 x5 x6 x7 x8}

Now let's discuss other type of random variables i.e.  Continuous random variables.

## Continuous Random Variables

A random variable that takes on an infinite number of values is known as a continuous random variable. Many physical systems (experiments) can produce infinite number of outputs in a finite time of observation. In such cases we use continuous random variables to define outputs of such systems.

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

## Examples of Continuous Random Variables

Example 1- A random variable that measures the time taken in completing a job, is continuous random variable, since there are infinite number of times (different times) to finish that job.

Example 2 - Noise voltage that is generated by an electronic amplifier has a continuous amplitude. Therefore sample space (S) and random variable (X) both are continuous.