Friday, December 11, 2015

0 probability ^ 0 density, again

See also the earlier post on 0 probability vs 0 density.

[[GregLawler]] P42 points out that for any continuous RV such as Z ~ N(0,1), Pr (Z = 1) = 0 i.e. zero point-probability mass. However the sum of many points Pr ( |Z| < 1 ) is not zero. It's around 68%. This is counterintuitive since we come from a background of discrete, rather than continuous, RV.

For a continuous RV, probability density is the more useful device than probability of an event. My imprecise definition is

prob_density at point (x=1) := Pr(X falling around 1 in a narrow strip of width dx)/dx

Intuitively and graphically, the strip's area gives the probability mass.

The sum of probabilities means integration , because we always add up the strips.

Q: So what's the meanings of zero density vs zero probability? This is tricky and important.

In discrete RV, zero probability always "impossible outcome" but in continuous RV, zero probability could mean either
A) zero density i.e. impossible outcome, or
B) positive density but a strip width of 0

Eg: if I randomly selects a tree in a park, Pr(height > 9KM) = 0. Case A. For Case B, Pr (height = exactly 2M)

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