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I recently started wondering if the average of a truly random sequence between numbers x and y has to be the average of x and y itself.

This little JavaScript function seems to prove it (uses Math.random() which should be a pretty good source of random numbers). The bigger precision is, the closer is the result to 0.5:

function averageOfRandom(precision) { // returns the average of `precision` random numbers between 0 and 1
  const sequence = [];
  for(let i = 0; i < precision; i++) { // make the sequence
    sequence[i] = Math.random();
  }

let totalSum = sequence.reduce((a, b) => a + b);


return totalSum / sequence.length; // average
}

Then, is a random sequence the only sequence that can have such trait without knowing its own length? Maybe a sequence simply alternating x and y is also good?

Ilya Maximov
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  • Welcome to MSE ^_^ I'm not entirely sure what you're asking... You are correct that the sequence $[x]$, $[x,y]$, $[x,y,x]$, $[x,y,x,y]$, $[x,y,x,y,x]$, ... will have averages that tend to $\frac{x+y}{2}$ in the limit. – HallaSurvivor Jan 09 '21 at 20:37

1 Answers1

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The notion "truly random sequence" is not a well-known rigorous concept.

However, an independent sequence of random variables, each uniformly distributed on $[x,y]$, does have mean converging to the midpoint $\frac{a+b}{2}$. This is (a special case of what is) known as the "law of large numbers".

Your guess is correct that there are many different sequences with this same limiting behavior.

GEdgar
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