Legitimate probability density functions

This lecture discusses two properties characterizing probability density functions (pdfs).

Not only any pdf satisfies these two properties, but also any function that satisfies them is a legitimate pdf.

Therefore, in order to determine whether a function is a valid pdf, we just need to verify that the two properties hold.

Table of contents

Properties of probability density functions
How to check that a pdf is valid
How to build valid pdfs
Solved exercises

Properties of probability density functions

The following proposition formally describes the two properties.

Proposition Let be a continuous random variable. Its probability density function, denoted by , satisfies the following two properties:

Non-negativity: for any ;
Integral over equals : .

Proof

Remember that, by the definition of a pdf, is such that [eq5] for any interval . Probabilities cannot be negative, therefore and [eq7] for any interval . But the above integral can be non-negative for all intervals only if the integrand function itself is non-negative, that is, if for all . This proves property 1 above (non-negativity).

Furthermore, the probability of a sure thing must be equal to . Since is a sure thing, thenwhich proves property 2 above (integral over equals ).

How to check that a pdf is valid

Any pdf must satisfy property 1 and 2 above. It can be demonstrated that also the converse holds: any function enjoying these properties is a pdf.

Proposition Let be a function satisfying the following two properties:

Non-negativity: for any ;
Integral over equals : .

Then, there exists a continuous random variable whose pdf is .

The practical implication is that we only need to verify that these two properties hold when we want to prove that a function is a valid pdf.

How to build valid pdfs

The proposition above also gives us a powerful method for constructing probability density functions.

Take any non-negative function (non-negative means that for any ).

If the integralexists and is finite and strictly positive, then define

Since is strictly positive, is non-negative and it satisfies Property 1.

The function also satisfies Property 2 because [eq20]

Thus, any non-negative function can be used to build a pdf if its integral over exists and is finite and strictly positive.

Example Define a function as follows: [eq21] How do we construct a pdf from ? First, we need to verify that is non-negative. But this is true because $x^{2}$ is always non-negative. Then, we need to check that the integral of over exists and is finite and strictly positive: [eq22] Having verified that exists and is finite and strictly positive, we can define [eq23] By the above proposition, is a legitimate pdf.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Consider the following function: [eq25]

where .

Prove that is a legitimate probability density function.

Solution

Since and the exponential function is strictly positive, for any , so the non-negativity property is satisfied. The integral property is also satisfied because [eq29]

Exercise 2

Define the function [eq30]

where and .

Prove that is a valid probability density function.

Solution

implies $frac{1}{u-l}>0$ , so for any and the non-negativity property is satisfied. The integral property is also satisfied because [eq33]

Exercise 3

Consider the function [eq34] where and is the Gamma function.

Determine whether is a valid probability density function.

Solution

Remember the definition of Gamma function: is obviously strictly positive for any , since is strictly positive and $x^{z-1}$ is strictly positive on the interval of integration (except at where it is ). Therefore, satisfies the non-negativity property because the four factors in the productare all non-negative on the interval .

The integral property is also satisfied because [eq43]

How to cite

Please cite as:

Taboga, Marco (2021). "Legitimate probability density functions", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/legitimate-probability-density-functions.