HW Geometric:
Read Ash pp. 51-2. Also Finite
Geom. / Geometric Series, p. 36. HW Finish Day 12
Read Ahead: Poisson Ash sec. 2-5,
also p. 36, the exponential series.
HWassigning
Day 14 : Poisson Ash p. 68, the non-Review problems:
#2
#3 (Poisson and binomial. compare)
4, 5, 6, 7
A. Suppose that on the average a certain
store gets 5 customers per hour. Consider now a two-hour period.
a) What is lambda for the 2-hour period?
b) What is the probability that the store
will
get 12 or fewer customers in the two hour period? (Use the table)
c) What is the probability that the store
will
get 13 or more customers in the two hour period?
d) What is the probability that the store
will
get exactly 13 customers in the two hour period?
B. Use Siegrist
Applet, Poisson Experiment. Experiment: Number of arrivals in
[0,t], r is the expected number per unit time.
Lambda is shown as "mean", bottom of N column.
Do the Store of problem A, 2 hour period, 5
customers per hour. (t=2, r=5) Run a few simulations. Read
off Distribution Table (right of graph) , Distribution column,s the
probability of 13 arrivals.
Compare with your answer to (Ad). It should
match, within .001.
Run r up and down, note the skewness of the
distribution
for small r, and the approach to normal for larger r.
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Geometric and Neg. Binomial Day 12.That's
all we did. Poisson next time..
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
in the whole interval (whole n trials).
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8 trials, 3 hits
Handout Wed: Tables & stuff.
Will go over the proof of the
binomial~Poisson connection (handout, or p. 64) in class.
Note the table gives the probability < x, just like a Normal table.
Siegrist
Applet, Poisson Experiment. Experiment: Number of arrivals in
[0,t], r is the expected number per unit time.
Lambda is shown as "mean", bottom of N column.
(lambda = rt).
The 'timeline" at the top is cut into 100 little bins; so we could
think of this as modeling Binomial, n=100, p = lambda/100.
Compare Poisson: lambda = 10 (r=10, t=1) to binomial, n = 100, p =
1/10 Siegrist
Applet, Binomial Timeline.
Distributions
The Geometric measured the waiting time
till
the first Binomial success.
There is a parallel variable measuring the
waiting time till the first Poisson hit. It's called the
Exponential,
but it's a continuous R.V. so we'll see it later.
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