Math 300 , Spring 2002, Day 14, W, Feb 27 Hit
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class
No class Friday. In lieu of class: A paragraph or so on
How organized data can be of use to the activist or individual
serving needs;
An example, either from a talk or workshop you attend, or some
data you wish had been available in same, or some other example.
Geometric and Neg. Binomial, see Day
13
Here is the Homework from
there also.
Reading: Ash pp. 36. pp. 51-2. Ahead, Poisson dist, sec
2-5, pp. 63-67
HW: A and B if you didn't do them already. Kristin says
Bb works out.
A. Remember Day 5: We can now prove ( nC1-nC2+nC3-nC4+.....+
nCn = 1). Hint: Let x = 1 and y = 1 and look at (x-y)n
This theorem, for P(A or B
or C....), was used in problem 7d, p. 52, due today.
B. Pascal's triangle: a) Complete the row for n = 8, using
the handout and the row for 7.
b) Use algebra/arithmetic to show nCk = (n-1)C(k-1) + (n-1)Ck.
Try for smallish numbers, n =6, k=4, etc. See if you can show
it in the general case, using algebra.
New, assigned today: (Same as on day 13 only cleaned up, clarified)
C. Geometric/Neg. Binomial: a) Give
formula for " Success happened on or before k'th" = p
+ qp + q2p + q3p + ...+ qk-1p =
? Use the Finite Geometric Series . Check that this plus P(no
successes on first k) =1. (this was done in class, just repeat
it.)
b) Use Interactive Probability (Bernoulli Trials:
Experiment: Number of Trials for k successes): to get the histogram for
the Geometric distribution (k=1). Copy, roughly, the distributions
when p =. 2 and p = .8, on the same scale. Note each probability
is q times the previous one.
c) Use Interactive Probability as in b, to do
Negative Binomial. With k = 2 (waiting till second success), run
the value for p back and forth (between .2 and 1) to see the distribution.
Repeat with k = 3, 4, 5. Write what the shape is like at the extremes
and how it changes.
Freund problem sheet:
13, 15, 17
Ash p. 53, 16 (a,b,c are straightforward.
There are several ways of getting d. The shortest is the trickiest.
For e, write down the favorable sequences and their probabilities.
Then you can use the Geometric Series.)
17. Make a tree of the fav's.
Monday we will begin the:
Poisson distribution
X =Number of "hits" in a fixed interval of
space or time (interval can be area or volume)
= Number of successes in Binomial,
with p small and n large.
Lambda = np = expected number of hits/successes.
Handout: Tables & stuff. Will
prove the binomial~Poisson connection .
Read :Ash sec. 2-5.
HW: Ash p. 68, the non-Review problems:
#2
#3 (Poisson and binomial. compare)
4, 5, 6, 7
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