Probability Density Functions – II

A commonly observed type of distribution among continuous variables is a uniform distribution. For a continuous random variable following a uniform distribution, the value of probability density is equal for all possible values. Let’s explore this distribution a little more.

Since all possible values are between 0 and 10, the area under the curve between 0 and 10 is equal to 1.

Clearly, this area is the area of a rectangle with length 10 and unknown height h. Hence, you can say that 10 * h = 1, which gives us h = 0.1. So, the value of the PDF for all values between 0 and 10 is 0.1.

The cumulative probability for X = 0.5 is equal to the area under the curve between X = 0, the lowest possible value, and X = 0.5.

This area = 0.1 * 0.5 = 0.05.

Now you must be wondering when to use PDFs and when to use CDFs. They are both good for continuous variables, but which one is used more in real-life analyses?

Well, PDFs are more commonly used in real life. The reason is that it is much easier to see patterns in PDFs as compared to CDFs. For example, here are the PDF and the CDF of a uniformly distributed continuous random variable:

The PDF clearly shows uniformity, as the probability density’s value remains constant for all possible values. However, the CDF does not show any trends that help you identify quickly that the variable is uniformly distributed.

Now, let’s look at the PDF and the CDF of a symmetrically distributed continuous random variable:

Again, it is clear that the symmetrical nature of the variable is much more apparent in the PDF than in the CDF.

Hence, generally, PDFs are used more commonly that CDFs.

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