The Probability Density Function is a function that gives us the probability distribution of a random variable for any value of it. To get the probability distribution at a point, you only have to solve the probability density function for that point.

The cumulative distribution function is used to describe the probability distribution of random variables. It can be used to describe the probability for a discrete, continuous or mixed variable. It is obtained by summing up the probability density function and getting the cumulative probability for a random variable.

The cumulative distribution function of a random variable to be calculated at a point x is represented as Fx(X). It is the probability that the random variable X will take a value less than or equal to x.

Consider the diagram shown below. The diagram shows the probability density function f(x), which gives us a rectangle between the points (a, b) when plotted. f(x) has a value of 1/(b-a).

Now consider a point c on the x-axis. This is the point you need to find the cumulative distribution function at. According to the definition, you need to find the total probability density function up to point c. This means that you have to find the area of the rectangle between points a and c.

Since the cumulative distribution function is the total probability density function up to a certain point x, it can be represented as the probability that the random variable X is less than or equal to x.

Figure 4: CDF representation

As you need to get the total PDF sum between two points, you can also represent the CDF as the integration of PDF between the points it has been calculated at. The formula depicted below shows the cumulative distribution function calculated between points (a, b) for the PDF Fx(x).

Source – simplilearn,com