Can you clarify the part highlighted in red? Do you mean specifically to generate N random numbers from the uniform distribution between, say, 0 and L?
If I'm writing code for this experiment, I can't generate a random number without telling the computer precisely which probability distribution to use (and the corresponding distribution parameters). Most computer programs, for instance, will allow one to generate a random number uniformly between [0, L], but then you will need to supply the value for the upper limit L. (Reverend Bayes, is that you?)
Randomness comes in many forms (e.g., normal, uniform, exponential, etc. ) and I don't think it is possible to generate random numbers without specifying which probability distribution we are working with.