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A friend and I were having a debate about randomness and at one point, I said that it was possible to have a collection of random processes which were not random when "put together." He disagreed.

So, I put the question here more concretely and with more detail.

Suppose I have a large number of random processes, is it possible for the collection of those processes to non-random and also, is it possible to have a part of that collection be not random?

Thanks for the help

Jeel Shah
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  • So you have a collection of functions $f_1,\dots,f_n$ and each of them output numbers with some random distribution ? And you're asking if the collection of random variables you could obtain from these functions would be not random ? You mean, correlated or something ? – davcha Nov 03 '14 at 12:16
  • Read "Fooled By Randomness" by Nassim Nicholas Taleb – Nick Nov 03 '14 at 12:39

1 Answers1

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You are correct. The key is using dependence. For example, let $X$ be distributed as a continuous uniform random variable on the interval $[0,1].$ Let $Y=1-X.$ Then define $Z=X+Y.$ Now $Z$ is a constant, but composed of two random components.

There are more practical examples. Imagine a closed-loop system where components move among several states randomly. The sum of all components is fixed and non-random, but the number in each state is a random variable. You can also have one or more states that are not random, satisfying your last version.

soakley
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  • Thanks for the practical example; that will be much more easier to explain. What fields of mathematics do you have to know to study randomness? – Jeel Shah Nov 04 '14 at 16:27
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    I only know the applied side - probability and probabilistic modeling, stochastic processes, statistics. – soakley Nov 04 '14 at 18:49