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I'm looking for a continuous random variable with the following properties

  • It is not bounded towards $+\infty$.
  • The expected value of the maximum of x-many draws out of that random variable has a closed-form solution.

The more standard and well-known it is, the better. I have no idea how to perform this search. The first property is immediately obvious. I've tried looking at Normal and Pareto, and compute

$$\int x F(j)^{x-1} F'(j) j dj$$

where $F(j)$ is the CDF of the RV, and $x$ denotes the number of draws. In both cases, this integral became quite messy (See here for a question regarding Pareto).

What is a recommended way of finding such a RV? Is there perhaps an obvious candidate?

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FooBar
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    Well, what common distributions do you know and what did you get? – user21820 Apr 15 '16 at 09:33
  • The simplest choice would be the uniform distribution over any interval you like. Very well behaved. Oops missed the fact that you want it unbounded. – tomi Apr 15 '16 at 09:36
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    @user21820 I've tried normal and Pareto so far. In both cases, expressions became tedious and I wasn't sure whether I was doing the right thing. With Pareto, "I" (Wolfram Alpha) was able to get an indefinite integral, but got stuck there (see: http://math.stackexchange.com/questions/1743535/expected-maximum-of-pareto) – FooBar Apr 15 '16 at 09:39
  • Try exponential distribution. By the way, the expectation on this site is that you describe own thoughts and work are in the question briefly, which I'm guessing is the reason for that downvote. – user21820 Apr 15 '16 at 09:40
  • @user21820 Thanks for the feedback - I've updated the question. Why do you think Exponential might work, experience of working with that particular distribution? I'd like to understand the methodology of getting there just as much as I appreciate an answer on this particular issue. – FooBar Apr 15 '16 at 09:52

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For a distribution $X$ with cumulative density function $F$, the probability that $n$ variables independently drawn from $X$ are all at most $r$ would be $F(r)^n$. That gives you the cumulative density function for the maximum of those $n$ variables, and from there you can get everything you want.

Now the exponential distribution is nice because it's using the exponential function, which remains like itself under plenty of operations, but many other distributions should work for your purpose too.

user21820
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Something else I was hinted at is to use the Frechet distribution, as it preserves maxima.

More precisely, if $X_i \sim Frechet(\alpha, s, m)$, then $\max \{X_1, \cdot X_n\} = Frechet(\alpha, n^\frac{1}{\alpha}s, m)$.

FooBar
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