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A chart of means of sets of pseudo random numbers:-

chart

The x axis is the size of the random set $X$, again chosen randomly between 20,000 and 1,024,000. The y axis is the calculated mean of that set. There are a million dots above, so a million means. Clearly $E(X) =127.5, (\frac{0 + 255}{2})$. Yet for finite sets, $\mu(X)$ is unlikely to be exactly 127.5.

The chart should allow (2D) interpolation for all random set sizes in the above range, but that's the bit I don't know how to do as a cumulative distribution function would have to be obtained.

Q$1$. Having this empirical chart, what is the probability that the mean of a 99,000 sized set is $\le 127$?

Q$1 \frac{1}{2}$. Is there an analytical way to answer this?

I've seen What is the expectation and sample mean of a random set? (unanswered).

Paul Uszak
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1 Answers1

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You can use the normal approximation for the total of your samples. The mean of the sum is the sum of the means and the variance is the sum of the variances. Your discrete uniform distribution has mean $127.5$ and variance $\frac {256^2-1}{12}$. For $N$ samples the mean of the mean will be $127.5$ and the standard deviation $\sqrt{\frac{256^2-1}{12N}}$. Now use the z-score table.

Ross Millikan
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