Took me a second to even parse what they were asking.
Recognize that the gap between the 5th and 6th order statistics is the spacing between the two middle order statistics. Use the known result that for n uniform points, the joint distribution of spacings is Dirichlet, and the marginal distribution of any single spacing is Beta(1, n). Then apply the specific parameters n=10 and the fact that the gap is a single spacing.
Pro tip: Connect the result to practical ML applications, such as understanding the distribution of distances between nearest neighbors in embedding spaces or the expected gap in quantile-based data splits, to show depth beyond the math.
Clarify that the 5th and 6th order statistics are the two middle values when the 10 points are sorted. The gap is the difference between them.
For n points uniformly distributed on [0,1], the spacings (including the endpoints) follow a Dirichlet(1,...,1) distribution. Each individual spacing (excluding endpoints) is Beta(1, n).
The gap between the 5th and 6th order statistics is the 5th spacing (if we consider spacings between consecutive order statistics). Since there are 10 points, there are 9 internal spacings, each Beta(1,10).
Thus, the gap follows a Beta(1,10) distribution. Its density is f(x)=10(1-x)^9 for 0<x<1. The expected gap is 1/11, and the variance is 9/((11)^2*12) = 9/1452 ≈ 0.0062.
Mention that the distribution is skewed, with most gaps small but occasional large gaps. This is relevant for understanding data sparsity in high-dimensional spaces or in quantile-based methods.
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