1. Take 12 random values, each uniform between 0 and 1. Add them up and subtract 6. Prove that the result is close to a random value drawn from the Normal distribution with mean zero and standard deviation 1.
Due to time limitations, I scanned my solutions.
Solution can be found here:
2. Invent a family of functions, each different, that look like those in Slide 3: they all have value 1 at x = 0; they all have zero derivative at x = 0; and they generally (not necessarily monotonically) decrease to zero at large x. Now multiply 10 of them together and graph the result near the origin (i.e., reproduce what Slide 3 was sketching).
The Matlab code is as follows:
syms x; y1 = 1/(1+x^2); y2 = 1/(1+x^3); y3 = 1/(1+x^4); y4 = 1/(1+x^5); y5 = 1/(1+x^6); y6 = 1/(1+x^7); y7 = 1/(1+x^8); y8 = 1/(1+x^9); y9 = 1/(1+x^10); y10 = 1/(1+x^11); y_mul = y1*y2*y3*y4*y5*y6*y7*y8*y9*y10; b = 2.0; ezplot(y1, [0,b]) hold on ezplot(y2, [0,b]) hold on ezplot(y3, [0,b]) hold on ezplot(y4, [0,b]) hold on ezplot(y5, [0,b]) hold on ezplot(y6, [0,b]) hold on ezplot(y7, [0,b]) hold on ezplot(y8, [0,b]) hold on ezplot(y9, [0,b]) hold on ezplot(y10, [0,b]) hold on h = ezplot(y_mul, [0,b]); set(h, 'Color', 'r'); % Make the last line red
3. For what value(s) of <math>\nu</math> does the Student distribution (Segment 8, Slide 4) have a convergent 1st and 2nd moment, but divergent 3rd and higher moments?
Class activity on this day was a continuation of the last problem set ..
You can refer to: Kai for our work.