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# 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 these two properties is a legitimate pdf.

## Properties of probability density functions

The following proposition formally describes the two properties.

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

1. Non-negativity: for any ;

2. Integral over equals : .

Proof

Remember that, by the definition of a pdf, is such that:for any interval . Probabilities cannot be negative, therefore and:for any interval . But the above integral can be non-negative for all intervals only if the integrand function itself is non-negative, i.e. 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, then:which proves property 2 above (integral over equals ).

## Identification of legitimate probability density functions

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:

1. Non-negativity: for any ;

2. Integral over equals : .

Then, there exists an absolutely continuous random variable whose pdf is .

This proposition 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: is strictly positive, thus is non-negative and it satisfies property 1. It also satisfies Property 2, because: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:How do we construct a pdf from ? First, we need to verify that is non-negative. But this is true, because is always non-negative. Then, we need to verify that the integral of over exists and is finite and strictly positive:Having verified that exists and is finite and strictly positive, we can define:By the above proposition, is a legitimate pdf.

## Solved exercises

Below you can find some exercises with explained solutions:

1. Exercise set 1 (identification of legitimate probability density functions).

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