Question

What is a Cumulative Distribution Function? How to derive c.d.f. from a probability density function?

Answer 1

The cumulative distribution function tells us - what is the probability that a random variable, say X, takes a value less than or equal to a specified value x? A C.D.F is usually denoted by F(x). 

                                  

F(x) lies between 0 and 1 since it denoted probability.

The probability that X lies in an interval, say x1 and x2 (both inclusive) is given by:

                                  

Let us understand visually what the above is actually saying.

                    

Let's say that the above is a probability density function of random variable X. F(x2) i.e. P(X <= x2) gives us the area under the curve ranging from negative infinity to x2. F(x1) i.e. P(X <= x1) gives the area under the curve ranging from negative infinity to x1. Now when we deduct F(x2) - F(x1), we get the area under the curve between x1 and x2. 

For the continous probability distribution of X, with pdf f(x) in the interval (a<=x<=b), the c.d.f is given as:

                    

 

A cumulative distribution function gives the probability of a random variable taking a value less than or equal to a specified value.