User:Tameem/Segment (10)

Problems

To Calculate

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:

Media:seg10_p1_tameem.pdf

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
 

Result:

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?

Solution:

Media:seg10_p3_tameem.pdf

Class Activity

Class activity on this day was a continuation of the last problem set ..

Teamed with with User:Noah, User:Kai, User:Trettels.

You can refer to: Kai for our work.