# Segment 10: Daniel Shepard

## Contents

### 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.

Solution:

First, the subtraction by 6 can be absorbed into the uniform distributions to give the equivalent problem of summing 12 random values each distributed uniformly on -1/2 to 1/2. The characteristic function of this sum of random values is given by




The plot below shows that this characteristic function is nearly identical to that of a normal distribution with 0 mean and standard deviation of 1, . The actual mean and standard deviation computed from the characteristic function are exactly 0 and 1 respectively.

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).

Solution:

I will use the family of sinc functions given by . The plot below shows the family of sinc functions for and the product of these functions, which appears to be approximately normally distributed.

3. For what value(s) of does the Student distribution (Segment 8, Slide 4) have a convergent 1st and 2nd moment, but divergent 3rd and higher moments?

Solution:

The first moment of the student distribution (assuming and without loss of generality) is given by




This integral converges for . The second moment is given by




For large , the integrand is approximately equal to . Thus, this integral converges for . The third moment is given by




For large , the integrand is approximately equal to . Thus, this integral converges for . Higher moments will require higher values of to converge, so the answer is .