Researches about theory
10_R. Distributions of the order statistics: look on the web for the most simple (but still rigorous) and clear derivations of the distributions, explaining in your own words the methods used.
11_R. Do a research about the general correlation coefficient for ranks and the most common indices that can be derived by it. Do one example of computation of these correlation coefficients for ranks.
Applications
10_A. Given a random variable, extract m samples of size n and plot the empirical distribution of its mean (histogram), the first and the last order statistics. Comment on what you see.
11_A. Discover a new important stochastic process by yourself! Consider the general scheme we have used so far to simulate some stochastic processes (such as the relative frequency of success in a sequence of trials, the sample mean and the random walk) and now add this new process to our process simulator.
Same scheme as previous program (random walk), except changing the way to compute the values of the paths at each time. Starting from value 0 at time 0, for each of m paths, at each new time compute N(i) = N(i-1) + Random step(i), for i = 1, …, n, where Random step(i) is now a Bernoulli random variable with success probability equal to λ * (1/n) (where λ is a user parameter, eg. 50, 100, …).
At time n (last time) and one (or more) other chosen inner time 1
Represent also the distributions of the following quantities (and any other quantity that you think of interest):
– Distance (time elapsed) of individual jumps from the origin
– Distance (time elapsed) between consecutive jumps (the so-called “holding times”)
Researches about applications
8_RA. Find out on the web what you have just generated in the previous application. Can you find out about all the well known distributions that “naturally arise” in this process ?
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