Cumulative Distribution Function: Overview, Questions, Preparation

A cumulative distribution function of a variable X, when evaluated at a value x, is said to be the probability at which x’s value is either equal or less than x. In simpler words, it is the cumulative probability of x at a given value. You can use it to analyse the probability of various values when exposed to specific conditions. 

The formula of Cumulative Distribution Function 

The cumulative distribution function can be given by the below formula:

Fx(x) = P(X≤x)

Here, the probability of the variable when its value is equal to or less than x, is given by X. It falls between the interval (a,b] (semi-closed) such that a is always greater than b. 

The probability of the variable within the given interval can be expressed as below:

P(a

If the random variable is continuous in nature, then the cumulative distribution function can be given as below:

 Fx(x) = ∫fx (t) dt such that the integral ranges from z to ∞

Properties of Cumulative Distribution Function 

For every cumulative distribution function, Fx is said to be right continuous and it does not decrease in value. 

If variable X is of continuous and random type, then the function fx is the same as Fx’s derivative, and it is given as below:

Applications of Cumulative Distribution Function 

The applications of CDF are given below:

The frequency analysis (cumulative) is used to compute frequency i.e. the number of times for which the value has occurred. 

Distribution function (empirical) makes use of CDF’s direct estimates of formal type. 

This topic is covered in Class XII and a question of at least 4 marks is asked in the exam, where you will need to calculate PDF or CDF or both. 

1. The PDF of a function is {2x   where 0 ≤ x ≤ 1 
                                               {0      otherwise  

X is the CRV (Continuous Random Variable). Therefore, find X’s expected value. 

Solution:

The expected value of X can be written as below

Expected value = ∫x fX (x)dx = ∫x (2) dx = ∫2x2dx

= ⅔ 

2. The random variable X has a PDF fx(x) = {cx   where  |x| ≤ 1 
                                                                           {0      otherwise 

Compute the value of the constant c.

Solution:

To compute c, we will have to use the below formula:

∫fX (u)du=1

1 = ∫(1,−1) cu2du

=(⅔)c

For the value of the function to be 1 c has to be 3/2.

3. Suppose that X∼Bernoulli(p). Calculate the estimated value of x.

Solution:

In a Bernoulli distribution, X’s range is (0,1), Px(0) = 1 - p,  and Px(1) = p. 

Therefore, the expected value = 0 x Px(0) + 1xPx(1) = 0 x (1-p) + 1xp = p

Q: What is cumulative frequency distribution?

Q: State some applications of CDF in the field of computers.

Q: What is a cumulative frequency polygon?

Q: What is the relation between PDF and CDF?

Q: Can a cumulative distribution be greater than 1?