Somebody rolls $n$ dice (each has $6$ sides, perfectly balanced) and tells me $k$ of them were $6$s. I'm trying to estimate $n$ based on $k$.

I think a good estimate for $n$ is $6k$ (or at least it makes sense to me intuitively) but I can't figure out:

1) how to prove that it isn't biased

2) how to compute the variance of this estimator.

Sorry if similar questions were asked before, it seems so simple that it must have been answered somewhere else but I couldn't find it.

  • 1
    $\begingroup$ the number of heads received is binomial, parameters p = 1/6 and n. The variance of the number of heads received is npq - so from that you should be able to work out the variance of the number of dice estimated - if you work from first principles of variance, what is he variance of 6X $\endgroup$ – Cato Feb 28 '17 at 12:46

Denoting $X_i$ as the number of times that number $i$ occurs by $n$ throws you are using $Y:=6X_6$ as estimator of parameter $n$.

$X_6$ has binomial distribution with parameters $p=\frac16$ and $n$.

It is well-known that $\mathbb EX_6=pn=\frac16n$ and $\text{Var }X_6=p(1-p)n=\frac5{36}n$, so that

  • $\mathbb EY=6\mathbb EX_6=n$ (i.e. $Y$ appears to be an unbiased estimate of $n$)
  • $\text{Var}Y=36\text{Var }X_6=5n$.

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