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.

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

:

.

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 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.

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

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 check 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

Define the

function

where

and

.

Prove that

is a valid probability density function.

implies

,

so

for any

and the non-negativity property is satisfied. The integral property is also

satisfied

because

Consider the

functionwhere

and

is the Gamma function.

Determine whether

is a valid 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

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.

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