Researches about theory
9_R. History and derivation of the normal distribution. Touch, at least, the following three i mportant perspectives, putting them into an historical context to understand how the idea developed:
1) as approximation of binomial (De Moivre)
2) as error curve (Gauss)
3) as limit of sum of independent r.v.’s (Laplace)
Applications
9_A_1. Create a simulation with graphics to convince yourself of the pointwise convergence of the empirical CDF to the theoretical distribution (Glivenko-Cantelli theorem). Use a simple random variable of your chooice for such a demonstration.
9_A_2. Generate sample paths of jump processes which at each time considered t = 1, …, n perform jumps computed as:
– σ sqrt(1/n) R(t)
where R(t) is a [-1,1] Rademacher random variable (https://en.wikipedia.org/wiki/Rademacher_distribution).
– σ sqrt(1/n) * Z(t), where Z(t) is a N(0,1) random variable (https://en.wikipedia.org/wiki/Normal_distribution)
and see what happens as n (simulation parameter) becomes larger.
[As before, at time n (last time) and one other chosen inner time 1
Researches about applications
7_RA Do a research about the random walk process and its properties. Compare your finding with your applications drawing your personal conclusions. Explain based on your exercise the beaviour of the distribution of the stochastic process (check out “Donsker’s invariance principle”). What are, in particular, its mean and variance at time n ?
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