Is a Random Number Generator Truly Uniformly Distributed?

In summary, random numbers can be randomly generated by taking a 4 digit number and its square. This can be used to get a random number between 0 and 1. You can also get a random number by taking a telephone book and selecting the last digit of every phone number.
  • #1
adityatatu
15
0
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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
The cardinality of the rationals is aleph_0 while that of the irrationals is aleph_1, I think !
 
  • #5
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.
 
  • #6
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…
 
  • #7
adityatatu said:
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. :smile:
 
  • #8
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.
 
  • #9
dear mathman;
can you explain more or give me a reference to study?
thanks in advanced.
:smile:
 
  • #10
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.
 
  • #12
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.
 
  • #13
Thanks a lot for the informations
 

What is a Random Number Generator?

A Random Number Generator (RNG) is a mathematical algorithm or device that generates a sequence of numbers that cannot be reasonably predicted better than by a random chance. It is commonly used in computer programming, statistics, and cryptography.

Why are Random Number Generators important?

Random Number Generators are important because they provide a reliable and unbiased source of random numbers, which are essential in many scientific and practical applications. These include simulations, statistical analysis, encryption, and games of chance.

What are the different types of Random Number Generators?

The three main types of Random Number Generators are:
1. Pseudo-Random Number Generators (PRNGs): These generate numbers that appear to be random, but are actually determined by a starting value called a seed.
2. True Random Number Generators (TRNGs): These generate numbers based on a physical process, such as atmospheric noise or radioactive decay, that is inherently random.
3. Hybrid Random Number Generators (HRNGs): These combine elements of both PRNGs and TRNGs to produce more secure and efficient random numbers.

What are some common applications of Random Number Generators?

Random Number Generators have a wide range of applications, including:
1. Statistical analysis and simulation in fields such as finance, engineering, and science.
2. Cryptography, where random numbers are used for key generation and encryption.
3. Gaming and gambling, where random numbers are used to determine outcomes.
4. Password and PIN generation for security purposes.
5. Art and music, where random numbers can be used to generate unique patterns and compositions.

How can the quality of a Random Number Generator be evaluated?

The quality of a Random Number Generator can be evaluated by several measures, such as:
1. Statistical tests: These check if the sequence of numbers produced by the RNG follows a uniform distribution and shows no discernible patterns.
2. Period length: This is the number of unique numbers that can be generated before the sequence repeats. A good RNG should have a long period length.
3. Entropy: This measures the randomness or unpredictability of the numbers generated. A higher entropy indicates a better RNG.
4. Speed and efficiency: A good RNG should be able to generate a large number of random numbers quickly and efficiently.
5. Cryptographic security: For applications that require high levels of security, the RNG should be evaluated against established cryptographic standards.

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