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

In summary, the conversation discusses the concept of probability and the misconception that the probability of a single event must always be 0. It is pointed out that for certain distributions, such as the uniform distribution, the probability of a single number can be non-zero. It is also mentioned that in infinite probability distributions, the events are defined as subsets of the set of all outcomes, with a measure assigned to each event to determine its probability.
  • #1
eMac
17
0
I was wondering why it is that the Pr(x=#)=0
 
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  • #2
? 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 probabilty 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.
 

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

1. What is the purpose of continuity correction in probability?

Continuity correction is used in probability to account for the discrepancy between continuous and discrete data. It is a way to approximate the probability of a discrete variable by using the probability of a continuous variable.

2. How is continuity correction applied in practice?

In practice, continuity correction is applied by adjusting the boundaries of a continuous distribution to align with the boundaries of a discrete distribution. This is done by adding or subtracting 0.5 from the endpoints of the interval.

3. Why is continuity correction necessary in some cases?

Continuity correction is necessary in some cases because the use of continuous distributions to approximate discrete distributions can lead to errors in probability calculations. It is particularly useful when dealing with small sample sizes or when the values being measured are discrete.

4. Can continuity correction be used with any type of probability distribution?

Yes, continuity correction can be used with any type of probability distribution, including normal, binomial, and Poisson distributions. It is a general technique that can be applied to any continuous distribution to approximate the probability of a discrete variable.

5. How does continuity correction affect the accuracy of probability calculations?

The use of continuity correction can improve the accuracy of probability calculations, particularly when dealing with small sample sizes or when the values being measured are discrete. It helps to reduce the errors that may occur when using continuous distributions to approximate discrete distributions.

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