Why Is Pr(X=#) Zero in Continuous Distributions?

[eMac]

I was wondering why it is that the Pr(x=#)=0

 

[HallsofIvy]

? As usual with one line questions, that makes no sense at all. Take a few lines to say what you are talking about. I can guess that "P(x= #)" means the probability that the random variable is a specific given number. But that makes no sense without saying what probability distribution you are talking about. And, indeed, for many distributions, what you are saying, that the probability of a singleton set is 0, is simply not true. For example, if my underlying set is {1, 2, 3, 4, 5, 6} and the probability distribution is the uniform distribution, then the probability of anyone number is 1/6, not 0.

I suspect you knew that but were talking about infinite probability distributions. But even then, it is not true that the probability of a single number must be 0. For example, I can define a probability distribution on the set of all positive integers such that [/itex]P(n)= 1/2^{n+1}[/itex]. The sum of all probabilities is a geometric series that sums to 1. Or I can define a probability distribution on [0, 1] such that P(1/2)= 1 and P(x)= 0 for all other numbers in [0, 1].

It is, however, impossible to have a uniform distribution on an infinite set with each outcome having non-zero probility because the infinite sum of a constant is not finite and so cannot be equal to 1, which is required for a probability distribution. Typically what is done is to define the "events" to be subsets of the set of all possible outcomes and define some "measure" of the set, say, length of an interval for one-dimensional problems and area for two-dimensional problems. Then the "probability" of an interval or area is its length or area divided by the length or area of the set of all outcomes. Of course, the "length" or "area" of an individual point is 0.