Will linear interpolation of a random value from one range to another 'preserve' randomness?

Question

I have generated a number z from a range (x,y) and want to map z to another range (p,q)

This can be achieved using linear interpolation as seen here: https://www.mathworks.com/matlabcentral/answers/379380-mapping-of-a-random-number-in-one-range-to-another-range

and here:

https://stackoverflow.com/questions/58773896/interpolation-in-java-mapping-a-random-number-from-one-range-to-another

The relationship I am trying to achieve is similar to this:

01, 02, 03, 04, 05, 06, 07, 08, 09, 10
25, 25, 25, 26, 26, 26, 27, 27, 27, XX

Where the x,y is the range 1-10, p,q is 25-27

If z=7, than the 'equivalent' in the second range would be roughly 27, however this would require one of the intervals to be rounded up or down so it's not totally accurate.

My question is, will linear interpolation achieve the above relationship, and if so, does the relationship 'preserve' the randomness?

For example, if I were to have rolled a die, does the second, interpolated value represent the same 'roll' so to speak, but in the context of a die with more sides?

Answer 1

If you need a deterministic function to map between the two intervals, then in general, as Ethan Bolker points out in his answer and attached comments, one cannot create a perfectly satisfactory map. It can only be done when the number of input values is an integer multiple of the number of output values.

If, however, you can use a probabilistic function, then it can be done, by mapping some input values to two or more output values, in a way that preserves the appropriate probabilities.

For example, suppose you want to map $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ to $\{25, 26, 27\}$ in a way that preserves the uniformity and also the ordering of values, as much as possible. That is, we want to define our mapping $f$ so that

• If $x < y$, then $f(x) \leq f(y)$
• If $x > y$, then $f(x) \geq f(y)$

Then, we would say $f(1) = f(2) = f(3) = 25$, $f(5) = f(6) = 26$, and $f(8) = f(9) = f(10) = 27$. In addition, we have the following random variables:

$$ f(4) = \begin{cases} 25 & \text{with probability $1/3$} \\ 26 & \text{with probability $2/3$} \end{cases} $$

and

$$ f(7) = \begin{cases} 26 & \text{with probability $2/3$} \\ 27 & \text{with probability $1/3$} \end{cases} $$

This should be straightforward to do in code.

Answer 2

The answer you have accepted to the linked stackoverflow question calls on the simple algebra for linear interpolation.

Linear interpolation from one range of integer values to another will almost always require rounding. Asking whether the rounded value represents "the same roll" is not a mathematical question. The answer will depend on context.

Rounding can't preserve "randomness". The chances of $1$, $2$ and $3$ on a three sided die are $1/3$ each. There is no way to map those to the range for a $5$ sided die that produces each answer with probability $1/5$.

You won't need rounding just when the number of integers in the domain is a divisor of the number of integers in the codomain - for example, converting a roll of a $6$ sided die to one of $12$ sides. Then you map $r$ to $2r$. You get all the even rolls with equal probability, but never the odd ones.