Derive expected inversions and mean distribution

Q: Derive expected inversions and mean distribution

This question evaluates understanding of permutation inversion statistics, computation of random-variable moments (expectation and variance), and the use of Monte Carlo simulation for empirical estimation, and sits in the Machine Learning domain with a strong probability and statistics component.

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Question

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Random permutation inversion statistics

Let π be a uniformly random permutation of length N. Let X be the number of inversions in π.

Compute the expected value E[X] .
Compute the variance Var(X) .

Monte Carlo experiment

Now suppose you run the following simulation:

Fix N = 200 .
Repeat T = 1000 trials:
- sample a fresh uniform random permutation of length N
- compute its inversion count X_t
compute the empirical mean \bar{X} = \frac{1}{T}\sum_{t=1}^T X_t and empirical variance across trials.

Answer the following:

What are the theoretical values of E[\bar{X}] and Var(\bar{X}) ?
What is the approximate distribution of \bar{X} for T=1000 (and why)?
If you repeat the entire simulation above 5 independent times (each time producing a mean \bar{X} ), what is a reasonable approximation for the typical range max(\bar{X}_1..\bar{X}_5) - min(\bar{X}_1..\bar{X}_5) ?

Derive expected inversions and mean distribution

Random permutation inversion statistics

Monte Carlo experiment

Solution

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Derive expected inversions and mean distribution

Overview

Random permutation inversion statistics

Monte Carlo experiment

Solution

Comments (0)