A more rigorous definition of a

probability density

function (from a

mathematical instead of a

statistical point of view--

Statistics is just a manipulation of probability):

First of all, in order to have a

density function, the random variable that it describes must be

continuous.

A random variable

X is continuous if its

probability distribution function, F(x) = P(X <= x) (the probability that a random variable

X is less than or equal to some

value of X, represented by x.) can be written as:

**
F(x) = from (-infinity, x) ∫ f(u) du**
for some integrable f:

**R** -> (0, infinity).

f is called the probability density function of

random variable X.

The density function of F is not prescribed uniquely by this integral, since two

integrable functions which take identical values except at some specific point

have the

same integrals. However, if F is differentiable at u, then we will normally set f(u) = F'(u).

So what does probability have to do with this? Remember, since X is a continuous random variable, it is just that: continuous. For example, if the RV

X is continuous (can take any value) between

0,5, the Probability that X = 3 is zero. There are infinitely many

values in (0,5), so a particular one has

probability 0.

However, one can find the probability that the value is between certain values, a and b by taking the integral:

**1) P(a <= X <= b) = (a to b)∫ f(x) dx**R such as the interval (more on this below).

Another property, mentioned above, is that:

**2) (-infinity to +infinity) ∫ f(x) dx = 1.**
*But...why does this characterize density functions?*
You had to ask. Let

*J* bet he collection of all

open intervals in

**R**.

*J* can be extended to a unique smallest

σ field *B* = σ(

*J*) which contains

*J*;

*B* is called the

Borel σ-field and contains

Borel Sets. B is a member of

*B*. Setting Px(B) = P(x member B), we can check that (

**R**,

*B*, Px) is a probability space. Secondly, suppose that f:R->

0, infinity (mapping onto the set of

real numbers) is integrable and (2).
So for any B in

*B*, we define

**P(B) = (over B) ∫ f(x) dx**
Then (

**R**,

*B*, Px) is a probability space and f is the density function of the

identity random variable X:

**R** ->

**R** given by X(x) = x for any x member

**R**.

Now, the

standard normal distribution is a perfectly good example of a

continuous random variable, but it is, despite what

statisticians want you to think, not the only one. The

exponential distribution is very useful, as well as the

gamma distribution,

Cauchy distribution, the

Beta distribution, and the

Weibull distribution.