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Show that the average of numbers given by a full cicle of a linear congruential generator with full lenght period is $\frac{1}{2}-\frac{1}{2m}$, whith $m$ as the module of the generator.

I began by using the definition of a linear congruential generator and the Hull & Dobell theorem, but I got lost in a cycling argument.

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  • $\begingroup$ You should define $m$. $\endgroup$ – Ross Millikan Aug 21 '20 at 23:03
  • $\begingroup$ Yeah, I'm sorry, $m$ is the module. $\endgroup$ – Davshock Aug 21 '20 at 23:40
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Hint: if the generator has full period, each possible value must show up once. What is the average of those?

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  • $\begingroup$ If $m$ is the module, it has $m$ different values, with the first one equals cero, so the average is $(m-1)/2$. $\endgroup$ – Davshock Aug 21 '20 at 23:48
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    $\begingroup$ I believe you are to divide once more by $m$ so the output is between $0$ and $1$. That gives just what you need. $\endgroup$ – Ross Millikan Aug 22 '20 at 0:09
  • $\begingroup$ Thank you very much for your help, I forgot that output. $\endgroup$ – Davshock Aug 22 '20 at 0:42

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