Legitimate probability density function

Probability Density Functions
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.

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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