How are the Real Numbers distributed?

In summary, the probability that a random variable X with domain all real numbers will take a value in the closed interval [a,b] depends on the distribution function of real numbers. The real numbers do not have an intrinsic distribution, but one can be defined based on desired characteristics such as a uniform distribution.
  • #1
HyperbolicMan
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Question: What is the probability that a random variable X with domain all real numbers will take a value in the closed interval [a,b]?

It seems to me that in order to answer this question you have to know how the real numbers are distributed. Given the appropriate distribution function, you can integrate it from a to b to find the probability.

Common sense says that the real numbers should have a constant distribution (e.g. P(x)=c for all x). However, the integral of any constant function from -[tex]\infty[/tex] to [tex]\infty[/tex] is not 1.

So how exactly are the real numbers distributed?
 
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  • #2
I am pretty sure that for any finite numbers a and b, the chance that x is in [a,b] is zero.

For any finite number a, the chance that x is in [a,infinity) is 50%. The same with (-infinity,a]
 
  • #3
HyperbolicMan said:
So how exactly are the real numbers distributed?
They are distributed how you want them to be. A set does not have an intrinsic distribution. For instance you may define a probability density function p(t) by p(t) = 1/2 for t in [1,3] and p(t)=0 otherwise.

If what you want is a uniform distribution, then no such thing can exist on the real numbers for the reason you mentioned (that if p(x) =c for all x, then [itex]\int_{-\infty}^\infty p(x) dx[/itex] is 0 or [itex]\infty[/itex] depending on whether c =0 or c>0, but never 1).
 

1. What is the definition of the Real Numbers?

The Real Numbers are a type of number that includes all rational and irrational numbers. They can be represented on a number line and have infinite decimal expansions.

2. How are the Real Numbers distributed on a number line?

The Real Numbers are distributed in a continuous manner on a number line, with no gaps or jumps. This means that between any two Real Numbers, there are infinitely many other Real Numbers.

3. Are there any patterns in the distribution of Real Numbers?

There are some patterns in the distribution of Real Numbers, such as the fact that there are an infinite number of whole numbers, but between any two whole numbers, there are infinitely many decimal numbers.

4. How are the Real Numbers different from other types of numbers?

The Real Numbers are different from other types of numbers, such as integers and fractions, because they include both rational and irrational numbers. They also have the unique property of being able to be represented on a number line.

5. Why are Real Numbers important in mathematics and science?

Real Numbers are important in mathematics and science because they are used to represent quantities and measurements in the physical world. They allow for precise and accurate calculations and are essential in many mathematical and scientific concepts and equations.

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