Total randomness, a definition

Question

First thing: is this the correct definition for total random integer generator?

An algorithm which for possible outputs $1,2,3 ... N$, there is a $\frac{1}{n} $chance of producing a single output in no predictable fashion.

With regards to my definition, my second question is why is it not possible to create something totally random? It is probably somewhere on SE, but I cannot find it anywhere. can somebody either give me a direct answer or reference to another SE question??

Edit: can this be expressed mathematically?

Answer 1

Most computer algorithm uses pseudorandom number generator to generate random numbers, but the problem is that if the seed (i.e. initial state) is the same, the number sequence generated will be the same. (thus that's not totally random)

But I feel it does not mean that we could not generate totally random numbers.

One possible way could be to read some totally un-predictable physical state (e.g. CPU temperatures into 10 decimal points).

Or a better way that I could think up of is to detect the quantum state (e.g. spin) of particles, which is random by nature.

Answer 2

Your definition is correct.

it's impossible to generate a random number. Because even if a human wants to generate some random number, the number they say is related to their thoughts. The computers generate a random number based on the time or the CPU state, and a formula which isn't a real random function(some numbers can't be generated at all).

The number that the computer generates, seems random to us because we don't know the exact time or the exact state of the CPU, and most importantly, we don't know what the formula is. So for us, the number the computer generates is totally random.(Possibilites of all numbers are equal) But if you know the formula and the state of the CPU and the time, you can easily predict what will be the number.

Answer 3

Sure it is. Take a Geiger counter and record the number of decay events per unit time of a radioactive sample. This will be totally random and Poisson distributed. Take data from a precise multimeter. This will be something close to a normal distribution. That sort of randomness arises all the time in nature and you don't need anything else but the ability to measure it.

As far as PRN or something akin to that goes, it's simply because the formulas used to calculate them are based on state machines. A good PRN is one with a good autocorrelation function -- very close to a delta function. But since everything is based on a state machine there is always going to be some value of the autocorrelation for shifted bits.

Answer 4

For the first question: you have defined a truly random uniform integer generator. (Integer probability distributions can also be non-uniform; it does not make them any less random.)

For the second question: it is impossible to create something truly random using a deterministic algorithm. Any of the usual "random" functions in computer programs, for example, actually are deterministic. Such an algorithm uses some internal state of the program to decide (in a completely reproducible way) what "random" value will be generated next.

On the other hand, we consider many events in quantum mechanics to be truly random. The events are not determined by the states of the objects involved.

Mathematically proving something to be random, on the other hand, is impossible. We can only rule out whatever deterministic patterns we can think of.

So it depends on exactly what you mean by "random" and what you mean by "impossible."