Exercises lecture 7

Exercise 1. Let’ s consider the famous Lorenz system, which is a set of three coupled nonlinear differential equations originally developed to model atmospheric convection. The equations are

The phase space of the Lorenz system exhibits chaotic behavior for certain values of the parameters σ, ρ, β

Start with
σ=10;
ρ=28;
β=8/3;
and explore other cases (e.g. with Manipulate)

Exercise 2 : The Central Limit Theorem (CLT) is a fundamental concept in statistics that states that the distribution of the sum (or average, or linear combination) of a large number of independent  random variables tends to be normally distributed, regardless of the original distribution of the variables. The exercise involves generating samples from a non - normal distribution, calculating their means, and then showing that the distribution of these means approaches a normal distribution as the sample size increases .

Generate a number ns of random variables from a uniform distribution in (0,1). Find the mean m. Then repeat this for ss times.
Then plot the histogram of the distribution of m.
Try with ss=10000 and ns=10, 100, 1000.
See how the distribution of m tends to a Gaussian for increasing ns.

You can try again with a different initial distribution, e.g. an exponential.