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My random variable is defined as $X=\max({x_1,...x_n})$, with $n$ very large. The $x_i$ are iid random variables following a Binomial distribution with with $k$ trials and success-probability $p$.

$k$ can be safely assumed to be large enough to approximate the distribution as a normal distribution.

Given the exponential decrease of the distribution of the $x_i$, for $n$ large enough X follows a Gumbel distribution, i.e. $P(X)=\frac{e^{\frac{\alpha -X}{\beta }-e^{\frac{\alpha -X}{\beta }}}}{\beta }$

Is there any known approximation (after a some research I believe there is no exact expression) of the parameters $\alpha$ and $\beta$ as a function of p and k (also an approximation valid only for large n would be ok)?

CupiDio
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  • In brief: if $(X_1, \dots, X_n)$ are a random sample of size $n$ from a $Binomial(k,p)$ parent, you seek the pmf of the largest order statistic. This can be obtained exactly ... no Gumbel or other approximations are required whatsoever, neither for small $n$ or large $n$. – wolfies Mar 02 '15 at 18:03
  • Apparently I wasn't searching with the right keywords... "extreme values distribution" didn't get me to anything, but "maximum order statistics" lead me directly to your answer: http://math.stackexchange.com/questions/276245/maximum-order-statistic-for-binomial-distribution to the same question!

    Can you suggest any reference where the topic is addressed anyway?

     – CupiDio Mar 02 '15 at 22:39
  • Well found ... I had forgotten about posting that solution. Sorry - I don't have any published reference to that solution (outside of stackexchange). – wolfies Mar 03 '15 at 14:49

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