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Functions of Random Numbers

Date: 08/26/98 at 16:59:26
From: Reg Reid
Subject: Random numbers

I don't know where to start thinking about this. If you add or multiply 
one random number by a non-random number, do you still get a random 
number? It seems that you would, but I can't prove it.

Date: 08/27/98 at 12:40:02
From: Doctor Peterson
Subject: Re: Random numbers

Hi, Reg. 

The answer depends partly on what you mean by random number. I assume 
you are thinking of an evenly distributed random number, that is, one 
that is equally likely to have any value within its range. But are you 
thinking of an integer, or a "real" number? And what do you think of as 
its range? That will make some difference.

Suppose you roll a die (a random number from 1 to 6) and then add 3 to 
it. Then you are equally likely to get any number from 4 to 10. So the 
result is random in that sense. But if you multiply it by 3 instead, 
you are equally likely to get any of 3, 6, 9, 12, 15, or 18. Some 
numbers between 3 and 18 will be missed, so although it is randomly 
distributed over those six values, it is not random in some senses.

If you are thinking of random numbers as generated by a computer, 
whether this makes a difference depends on what you are doing with the 
number. If you are generating random numbers in some range by getting 
a random number and then doubling it, the resulting numbers would be 
random enough for some purposes; but if you use them to sample some 
numbers and determine what percentage of them are even, you would think 
that all numbers are even! And if you used them to select math problems 
from a list to ask someone, you would find that you never asked the 
odd-numbered questions.

On the other hand, your random number may be given to you as a real 
number between 0 and 1. If you double it, you will get a random number 
between 0 and 2. What I've just described will still happen, but it 
will be hidden deep in the last digits of a decimal, and for most 
purposes you could ignore it.

So my answer is that the result is still "random," but only if you are 
aware of the restrictions you have made on the values the random 
number can have, and don't do anything that will be affected by it.

Does that help?

- Doctor Peterson, The Math Forum
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Associated Topics:
High School Probability

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