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What is more, one can also derive a strong finite-sample bound on this convergence

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

1. (Expansion in normal directions)

In this exercise X  and Z are two independent absolutely continuous random variables defined on the same probability space (Ω, A , P ) mapping to R. We assume that Z follows the standard Normal distribution (which I will introduce in class soon), whose density is

(b) (What do you expect the limit as ε → 0 to be? [Hint: Convolution]

2. (Order statistics, Part 2)

Consider three random variables X1, X2, X3, which are iid with uniform distribution on [0, 1]. Let X[1]  be  the  smallest,  X[2]  be  the  middle,  and  X[3]  be  the  largest  random  draw,  i.e.,  the  order statistics.

(a) (20 points) Find the joint density of the random variable

(c) Extrapolate your finding to n elements, where Zn

3. (Empirical distribution, Part 2)

Consider again the empirical distribution function from PS3,

What is more, one can also derive a strong “finite-sample” bound on this convergence – that is, for fixed n and ε > 0, an upper bound on the probability,

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