Assume $Z$ is a Binomial random variable with a distribution function $$ f(z)=\binom{n}{z}p^z(1-p)^{n-z} $$ What is the distribution of the random variable $Z/n$? Does it have a name? Any close form expression of the distribution of $Z/n$ will be highly appreciated.

  • $\begingroup$ It is a sample estimate of the proportion p, the maximum likelihood estimate.. $\endgroup$ – Michael R. Chernick Mar 14 '18 at 23:55
  • $\begingroup$ If Z is the number of "successes" out of n trials, then Z/n is the proportion of successes out of n trials. It has the exact same distribution as Z, just for arguments that are 0/n,1/n,2/n,...,n/n rather than 0,1,2,...,n. $\endgroup$ – Zahava Kor Mar 15 '18 at 1:09

We can get a closed form for the pmf but it's not very exciting. Let $W = Z/n$ and let $f_W$ and $f_Z$ be the pmfs of $W$ and $Z$ respectively. Since $$ f_W(w) = P(W = w) = P(Z/n = w) = P(Z = nw) = f_Z(nw), $$ we have $$ f_W(w) = f_Z(nw) = {n \choose nw}p^{nw}(1-p)^{n-nw} $$ supported on $w \in \{0, \frac 1n, \frac 2n, \dots, 1\}$ or equivalently $nw \in \{0, 1, \dots, n\}$.

I don't know if I've ever heard this given a canonical name but I think that's because the binomial is the fundamental aspect, not the scaling, and that can usually be made clear from context.

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