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I want to be able to do generalize this:

n = 3;
d[n_] := OrderDistribution[{UniformDistribution[], n}, Range[n]];
Probability[ x2/2 == Min[x1/1, x2/2, x3/3], {x1, x2, x3} \[Distributed] d[n]]

The goal is to be able to plug in k to get

$$P_{k,n} = P\left(\frac{x_k}{k}=\min{\left(\frac{x_i}{i}\right)} | x_i \textrm{ are the order statistics of }U_i, i =1,...n\right) $$

Basically, I want the probability that the $k_{th} $ order statistic divided by $ k$ is the minimum of all of the order statistics when divided by their index.

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1 Answer 1

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Here's one possibility that defines the random variables in the array a and then maps to find all the individual probabilities.

n = 3;
d[n_] := OrderDistribution[{UniformDistribution[], n}, Range[n]];
a = Array[x, n];
Probability[a[[#]]/# == Min[a/Range[n]], a \[Distributed] d[n]] & /@ Range[n]

{4/9, 2/9, 1/3}

Notice that the probabilities total to 1.

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