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.
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:
Non-negativity: for any ;
Integral over equals : .
Remember that, by the definition of a pdf, is such thatfor any interval . Probabilities cannot be negative, therefore andfor 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 ).
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 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 becauseThus, 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 defineBy the above proposition, is a legitimate pdf.
Below you can find some exercises with explained solutions.
Consider the following function:
where . Prove that is a legitimate probability density function.
Since and the exponential function is strictly positive, for any , so the non-negativity property is satisfied. The integral property is also satisfied because
Consider the following function:
where and . Prove that is a legitimate probability density function.
implies , so for any and the non-negativity property is satisfied. The integral property is also satisfied because
Consider the following function:where and is the Gamma function. Prove that is a legitimate probability density function.
Remember the definition of Gamma function: is obviously strictly positive for any , since is strictly positive and 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
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