# Maximum order statistic for Binomial distribution

Let $X_i$, $1\le i\le t$, be $t$ independent random variables with Binomial distribution $B(n,\frac1t)$.

I would like to find the distribution of $X_{Max}=\max_{i=1}^t(X_i)$

Note that this is the specific case where the probability of success is equal to the reciprocal of the number of variables I am taking the maximum over.

I expect that the answer will be too complex to use directly, so an approximation would be good to have. In fact just having an approximate formula for $\mathbb{E}(X_{Max})$ and $\mathbb{Var}(X_{Max})$ would be sufficient, and I am most interested in the case where $n\gg t\gg1$.

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Despite considerable upvoted interest in the question itself, ... this question has remained unanswered for over 4 months.

We are given $X$ ~ Binomial$(n,p)$ with pmf $f(x)$:

Let $(X_1, ..., X_t)$ denote a random sample of size $t$ drawn on $X$. Then, the pmf of the $t$-th order statistic (i.e. the pmf of the sample maximum), denoted say $g(x)$, is given by:

where OrderStat[t, f, samplesize] is a mathStatica function that finds the pmf of the $t$-th order statistic, and where the Hypergeometric2F1 is described here . The domain of support is, of course:

domain[g] = {x, 0, n}

This solution, for general probability parameter $p$, nests the special case the original poster wishes to solve for, namely where $p=\frac{1}{t}$. So simply plug in: $p=\frac{1}{t}$.

Normally, of course, as the sample size (here $t$) increases, the distribution of the sample maximum shifts out to the right. However, in this particular set-up, because $p=\frac{1}{t}$, the opposite happens ... the distribution of the sample maximum shifts to the left as $t$ increases, as the following plot illustrates:

[I have also 'checked' the above solution using Monte Carlo methods: all seems fine. ]

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