Generating random number in sorted order

Question

I want to generate random number in sorted order. I wrote below code:

void CreateSortedNode(pNode head)
{
    int size = 10, last = 0;
    pNode temp;
    while(         


        

Answer 1

Suppose you wanted to generate just three random numbers, x, y, and z so that they are in sorted order x <= y <= z. You will place these in some C++ container, which I'll just denote as a list like D = [x, y, z], so we can also say that x is component 0 of D, or D_0 and so on.

For any sequential algorithm that first draws a random value for x, let's say it comes up with 2.5, then this tells us some information about what y has to be, Namely, y >= 2.5.

So, conditional on the value of x, your desired random number algorithm has to satisfy the property that p(y >= x | x) = 1. If the distribution you are drawing from is anything like a common distribution, like uniform or Guassian, then it's clear to see that usually p(y >= x) would be some other expression involving the density for that distribution. (In fact, only a pathological distribution like a Dirac Delta at "infinity" could be independent, and would be nonsense for your application.)

So what we can speculate with great confidence is that p(y >= t | x) for various values of t is not equal to p(y >= t). That's the definition for dependent random variables. So now you know that the random variable y (second in your eventual list) is not statistically independent of x.

Another way to state it is that in your output data D, the components of D are not statistically independent observations. And in fact they must be positively correlated since if we learn that x is bigger than we thought, we also automatically learn that y is bigger than or equal to what we thought.

In this sense, a sequential algorithm that provides this kind of output is an example of a Markov Chain. The probability distribution of a given number in the sequence is conditionally dependent on the previous number.

If you really want a Markov Chain like that (I suspect that you don't), then you could instead draw a first number at random (for x) and then draw positive deltas, which you will add to each successive number, like this:

1. Draw a value for x, say 2.5
2. Draw a strictly positive value for y-x, say 13.7, so y is 2.5 + 13.7 = 16.2
3. Draw a strictly positive value for z-y, say 0.001, so z is 16.201
4. and so on...

You just have to acknowledge that the components of your result are not statistically independent, and so you cannot use them in an application that relies on statistical independence assumptions.