Continuous random variable: Zero probablity

[FactChecker]

I trust you mean PDF, not CDF.

Good point. Thanks. I worded my statement badly. I meant that the CDF is continuous, implying that the probability of any single exact resulting value is zero. -- Not that the cumulative probability is zero.

 

[FactChecker]

We can suppose such a thing can happen, but If we suppose that you (or Nature) can pick an exact mathematical point from a continuous distribution then we have made an assumption about physics.

Good point. In fact, I may have seen somewhere that in quantum theory time is in fact quantized, so location on an X-axis may also be quantized. I don't know enough to comment more than that. Even if that is true, I think that I would accept the approximation of the discrete physics with a continuous model for the purpose of ignoring any quantization of time-space.

I agree that the following physical situations are different:

1) Nature cannot select an exact result from a continuous probability distribution.

2) Nature can select an exact result from a continuous probability distribution, but we cannot measure what nature has done exactly.

So the fact we cannot measure an exact result from an experiment doesn't tell us whether 1) or 2) is the case.

My point about the mathematical theory of probability is that it does not assert we can do such an experiment with a dart. - i.e. it does not assert that 2) is the case.

I have to agree. At the finest level of detail, we may never know the answer. I will have to resign myself to the realization that, at the quantum level, the continuous CDF may not be possible. It may be an approximation.

 

[Joseph M. Zias]

Consider a block of wood whose linear density you know, say 100 g per cm. You acquire mass by spanning a distance. As that distance gets smaller so does the mass acquired, a span the thickness of a thin paper would be very small. In the limit as the span approaches zero you would of course have zero mass. The same effect is seen in spectrum analysis. As the bandwidth gets narrower the energy measured gets less, if you had a bandwidth of zero - a single frequency - you would have zero energy.

 

[sysprog]

Calling it zero probability is linguistically misleading -- only the impossible has zero probability -- calling something possible is the same as saying it has more than zero probability. The probability that 1 = 2 is 0. The probability that a number to be chosen from all real numbers will be 1.2 is > 0.