What is a Probabilty Density Function?

[Mr Davis 97]

What really is a probability density function for continuous random variables? I know that the probability for a single value occurring in a continuous probability distribution is so infinitesimal that it is considered 0, which is why we use the cumulative distribution function that is the the integral of the PDF from -∞ to some number x. However, if any single value in the PDF is 0, then how to we get a density curve and how are we able to integrate the PDF if all of its values (probabilities) are zero?

 

[pwsnafu]

if all of its values (probabilities) are zero?

Why do you think this is possible?

Hint: write down the definition of density function. It contradicts the part I quoted. Where?

 

[Stephen Tashi]

What really is a probability density function for continuous random variables?

You can equally well ask: what is a mass density function for physical object? For example if a rod lying along the x-axis has a variable mass density, we can give it a mass density function that is a function of x. There is a mass density "at point x", but the mass "at point x" is zero. A probability mass density function is no more and no less mysterious than a physical mass density function.

It is not physically possible to "take a point" from the rod and put it in a sample dish. It's also not physically possible to take a random sample from a continuous probability distribution.

 

[mathman]

In simplest terms, a probability density function is the derivative of a (nice) probability distribution function.

Specifically, "nice" functions are absolutely continuous.

 

[Stephen Tashi]

. However, if any single value in the PDF is 0,

To repeat pwnsnafu's comment. the values of a PDF f(x) need not be zero. You are saying "any single value in the PDF" when you mean "the probability of a single value of x computed by using the PDF". If f(x) is a probability density then f(x) is not equal to "the probability that the outcome is x". Instead, f(x) is equal to the probability density at x.

By analogy, the density of an object can be 1 gram per cubic centimeter at a point (x,y,z) without claiming that there is any mass "at" the point (x,y,z).

then how to we get a density curve

How do you get a mass density function if the mass of each point is zero? You take a limit of the mass per unit volume of sequence of volumes that shrink around the point. The probability density at x can be found by taking the limit of ( the probability of the event [x, x+ h]) / h as h approaches zero. This amounts to taking the derivative of the cumulative distribution and evaluating the derivative at x.

and how are we able to integrate the PDF if all of its values (probabilities) are zero?

As noted, above, the values of a PDF are not all zero.