Is a Random Number Generator Truly Uniformly Distributed?

[adityatatu]

Can somebody prove the following:

A Random number which (truly) is uniformly distributed on [0 1] (or in fact any continuous random variable) is irrational with probability 1.

 

[HallsofIvy]

In order for that problem to be "well-defined", you have to specify the measure. Assuming that the measure is the standard Lebesque measure, the set of all irrational numbers between 0 and 1 has measure 1 and the set of all rational numbers has measure 0.

 

[somy]

I don" know the prove! but I know an interesting way to get random numbers.
just take a 4 digit number. get its square. keep two first digits away. then get the 4 next number az the next step iteration.
some initial values dosent work. can you tell them?!

for example:
1-0.3265
2-3265
3-3265^2=10660225
4-10_6602_25
5-6602 ----0.6602
6_ go to step 1.

 

[Gokul43201]

The cardinality of the rationals is aleph_0 while that of the irrationals is aleph_1, I think !

 

[mathman]

It is easy to prove that the measure of the rationals is 0, because of countability. Make a list of rational numbers. Cover the nth member of the list (symmetrically) by an interval of length x/2ⁿ. All the rational numbers are then covered by a set of measure less than x, since the union of these intervals has measure less than the sum of the individual measures. Since x can be made arbitrarily small, the measure of the rationals is 0.

 

[h2]

Another way to get random numbers:
Get ur telephone book, open it anywhere, choose a column, and pick the LAST digit in every phone number from the column…

 

[somy]

Can somebody prove the following:

A Random number which (truly) is uniformly distributed on [0 1] (or in fact any continuous random variable) is irrational with probability 1.

I have a question:
What does random number exactly mean?
I have this question since I hear the word!
can you help me?
Thanks in advanced.

 

[mathman]

You need to first understand the concept of a random variable. It is a variable which has a value determined on the basis of a probability distribution. A random number is a random variable with a probability distribution uniform between 0 and 1.

 

[somy]

dear mathman;
can you explain more or give me a reference to study?
thanks in advanced.

 

[mathman]

I am sorry to say that I haven't looked at any recent material on the subject. Feller's 2 volume "An Introduction to Probability ..." is a very good, but old, text. You can try probability theory with google.

 

[Gokul43201]

Mathworld is a good resource : http://mathworld.wolfram.com/RandomNumber.html

 

[mathman]

The wolfram reference is a good description for those using random numbers. However, you need to look at a good basic probability textbook to understand the mathematical foundations behind the concept of random variable.

 

[somy]

Thanks a lot for the informations