1. Show that the Uniform distribution U(0,1) over the unit interval [0,1] is not a stable distribution. Define a random variable Q(n) which is a sum of n random variables x_1, x_2, ..., x_n, each of which are sampled from U(0,1). Now generate many values of Q(n) and see how it is distributed for different n.
For n=1, Q(1) will of course have the distribution U(0,1).
But what about n=2 ? Is it still U(0,1).
How about n=3 or n=5 or n=10 ?
Can you show roughly how large n has to be before it starts resembling a Gaussian distribution ?
2. Can you do a convolution of two uniform distributions and show that the distribution of Q(2) will be triangular ?
3. Suppose instead of U(0,1), you chose the elements of Q(n) from a Gaussian distribution with mean 0 and variance 1. What will be the distribution of Q(n) for various values of n ?
4. Suppose you chose the elements of Q(n) from the Cauchy distribution P(x) = 1/(pi*(1+x^2)). Will Q(n) be distributed as a Gaussian for large n ?
5. Show that if you have a random variable L(n) defined as the product of n random variables each of which can be either 2 or 1/2 with equal probability (=0.5), then for large n the distribution followed by L(n) will be log-normal.
6. There are other ways of generating a log-normal distribution. Consider the sequential broken stick model. Take a stick which is of unit length and break it in half. Then choose one of the two resulting pieces at random and again break it in half. You now have 3 pieces. Again choose one and break it in half. Continue doing this until you have n pieces. Show that the distribution of the lengths of these n pieces will be log-normal if n is sufficiently large.