Math 300 , Spring 2008, Day 21, F, March 14 Hit reload...

Added end of term: use it next time I teach it
I know the course is over but I can't stop teaching, in this case what I just learned at sci colloq today!

Remember we learned that the Poisson distribution has both mean and variance equal (so the ratio is 1). I said it was interesting but I didn't know what to do with that...Now I do:

Turns out that the variance-to-mean ratio can be used to examine a distribution of things in an interval of time or space as to how clustered or "regular" they are! Also called the coefficient of dispersion. .http://en.wikipedia.org/wiki/Coefficient_of_dispersion. So, think of a short interval, with expected value say 3 observations in the interval.

Look at a long interval which is made up of a sequence of short intervals of this length, and gather data, finding the distribution of "hits" in the short intervals.

If the distribution of hits is *random*, the distribution is Poisson, that is, P(X hits in a short interval =x)follows the Poisson dist. with mean 3. The variance-to-mean ratio is 1.

If each interval gets exactly 3 observations, that would be very *regular* (uniformish, but it's not really the "uniform distribution"), and the variance (of observations in an interval) would be 0. The variance-to-mean ratio would be 0.

If the data is *More clustered* than we would see in Poisson, the short intervals would be more likely to have 0's and (say) 6's than 3's, and have a greater spread (variance) than Poisson. The variance-to-mean ratio would be > 1.

So the Variance-to-mean ratio goes from 0 (completely regular) to 1 (random) and greater than 1 (more clustered than random).

Ecologists use this all the time! (Prof. Vawter now tells me.) Cross-section a field, count the number of dandelion plants in each square. Are dandelion plants clustered (as if the seeds had been dropped close to the parent plant), random (my guess for dandelions, considering the seeds are blown all around), or regular (planted in a more or less neat grid; like apple trees in an orchard--or if each plant does something to kill other plants within its personal space).

So I found out about this today in Sci. Colloq. when Allison Inga used it to show that the distribution of egg follicles in the mouse ovary is clustered, not random as every scientist up to now has been assuming. (She had to make microslices of mouse ovaries and count all the follicles--many hundreds total--in little grids on the slices. That's why I prefer doing the math to doing the science...;-) Beautiful work.

Happy whatever,
srs

Midterm due  Wed. Mar 26, Day 23.
HW:  A.  Application of covariance/correlation (Some may find youselves working in finance. With luck you will all someday have money to invest)  M&M 5e, Investment portfolios.  Example 4.28, p. 304, problems 4.83, 84, 85 on p.311-12.  (Coming into the present recession, the traditional negative correlations aren't holding up--bonds aren't moving opposite to stocks, because nobody trusts them any more because of the mortgage crisis.  Hard times.)
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HW  Read Ash pp. 95-6.  Graph f(x) on p. 95 and g(x) on p. 96 --piecewise-defined functions (Prep. for  Ash Ch. 4, Next.)
Have a great break.
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  Questions??
Variance of Hypergeometric, completed, handout . 
  Cov(X_i, X_j) is negative!  discussion    Day 20

Variances of Binomial and Poisson:
Binomial X : mean np, variance npq.  Standard deviation is square root of variance. (same units as X)
Keep p constant, let n increase.  Range of possible values is 0--n.  Mean stays in same place in this scale (p of the way from 0 to n)  But the standard deviation grows not like n but like square root of n; bulk of probability takes up less and less of total possible range.  See binomial applet, let n go from 5 to 60.  Try different p's.  Bar below axis shows mean, and 1 s.d. either side.   Mean and s.d. are given numerically at the bottom of the "Distribution" column.
Poisson: Mean lambda, variance lambda. Standard deviation is square root of variance.  In the Poisson applet, lambda = ratextime. Increase lambda by increasing  rate per unit, and/or increase number of units of time.  Spread increases like square root of lambda. It's a little harder to grasp visually for the Poisson than for the binomial, because the Poisson graph is re-calibrated to keep the distribution spread looking about the same.  Look at the width of the mean+s.d.bar compared to the distance of the mean above 0. (Note the y axis doesn't cut the x axis at 0 for larger lambdas.)   Visualize for lambda = 1 (s.d. = 1) and increase to lambda = 100 (s.d. = 10)

Moral: the variability grows like the square root of the number of trials; not nearly as fast as the possible outcomes.  This allows us to find predictability in "averages".

More time?  Two-asset portfolios, aX+bY, a+b=1; lower risk by choosing X and Y with negative correlation.
Moore applet: http://bcs.whfreeman.com/ips5e/   Two-asset portfolios
   Mousing on the curve will show the mean and s.d at that point.  It doesn't tell you the a's along the curve.  But you can guess and refine using a smaller range of a's +New Plot to narrow down.    Try X mean 20, sd 5, Y mean 15, sd 2,  r= -.3  The safest mix is a about .2...1/5 X; but the mean return goes down to 16.  Note that it's both safer and higher return than all Y.
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Quiz Friday after break: Closed book, Expected value stuff, chosen from: