Segment 10...The Central Limit Theorem
Problems from Segment 10. The Central Limit Theorem
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.
Let be random variables, each uniformly distributed on 0 and 1.
and (use wolfram alpha or ) .
Let . Then by the Central Limit Theorem,
So for ,
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).
For , define . These all have the properties desired above, i.e. and , and here they are plotted out in MATLAB:
and here is the plot of their product:
(let me know if you know a more efficient way to evaluate the functions, many thanks)
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?
Staring at the pdf for the Student distribution, notice that . So for and :
and the higher moments will be proportional to the integrals of polynomials over the real line, which are also divergent.
To Think About
1. A distribution with moments as in problem 3 above has a well-defined mean and variance. Does the CLT hold for the sum of RVs from such a distribution? If not, what goes wrong in the proof? Is the mean of the sum equal to the sum of the individual means? What about the variance of the sum? What, qualitatively, does the distribution of the sum of a bunch of them look like?
2. Give an explanation of Bessel's correction in the last expression on slide 5. If, as we see, the MAP calculation gives the factor 1/N, why would one ever want to use 1/(N-1) instead? (There are various wiki and stackoverflow pages on this. See if they make sense to you!)
Just for fun
A fun problem that ties in to 'To Calculate' 1 above and problem 6 from the Probability Blitz:
1. What is the expected number of Uniform[0,1] draws you need to add up before the sum exceeds 1? Prove your answer analytically and confirm it by simulation.
We got the formula for the PDF from the Irwin-Hall wikipedia page.
In Maple, we plotted this function for N=12 and subtracted it from a normal centered at 6.