Saturday, September 14, 2013

Prob Density Func - Pr(X=any valid value) == 0 always?

Given a simple uniform distribution over [0,10], we get a paradox that Pr (X = 3) = 0.

http://mathinsight.org/probability_density_function_idea explains it, but Here's the way I see it.

Say I have a correctly programmed computer (a "noisegen"). Its output is a floating point number, with as much precision as you want, say 99999 deciman points, perhaps using 1GB of memory to represent a single output number. Given this much precision, the chance of getting exactly 3.0 is virtually zero. In the limit, when we forget the computer and use our limitless brain instead, the precision can be infinite, and the chance of getting an exact 3.0 approaches zero.

http://mathinsight.org/probability_density_function_idea explains that when the delta_x region is infinitesimal and becomes dx, f(3.0) dx == 0 even though f(3.0) != 0.

Our f(x) is the rate-of-growth of the cummulative distribution function F(x). A gradient f(3.0) = 0 has some meaning but it doesn't mean there's a zero chance of getting a 3.0. In fact, due to continuous nature of this random variable, there's zero chance of getting 5, or getting 0.6 or getting a pi, but the pdf values at these points aren't 0.

What's the real meaning when we see the prob density func f(), at the 3.0 point is, f(3.0) = 0.1? Very loosely, it gives the likelihood of receiving a value around 3.0. For our uniform distribution, f(3.0) = f(2.170) = f(sqrt(2)) = 0.1, a constant.

The right way to use the pdf is Pr(X in [3,4] region) = integral over [3,4] f(x)dx. We should never ask the pdf "what's the probability of hitting this value", but rather "what's the prob of hitting this interval"

The nonsensical Pr(X = 3) is interpeted as "integral over [3,3] f(x)dx". Given upper bound = lower bound, this definite integral evaluate to zero.


As a footnote, however powerful, our computer is still unable to generate most irrational numbers. Some of them have no "representation" like pi/5 or e/3 or sqrt(2), so I don't even know how to specify their position on the [0,1] interval. I feel the form-less irrational numbers far outnumber rational numbers. They are like the invisible things between 2 rational numbers. Sure between any 2 rationals you can find another rational, but within the new "gap" there will be countless form-less irrationals.

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