The function f(x), illustrated in the graph below (which represents a HTML bell shape curve), is that of the Standard Normal probability density function or PDF. The Standard Normal PDF has mean=0 and standard deviation=1.

            * :    |    : *
          *   :    |    :   *
        *     :    |    :     *
      *       :    |    :       *
    *         :    |    :         *
***           :    |    :           ***
    2      -1 a    0    b 1       2

The area under any continuous PDF is 1, as indicated in the following equation -

∫ [ (e-0.5x²) / (√2π) ]dx = 1

The area sectiond by ' : ' represents the probability that the random variable X assumes a value between X=a and X=b
OR P(a<X<b).
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.

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