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).
|___|_x_|___|___|__x|___|x__|___|
8 trials, 3 hits
Handout: Tables & stuff. Will
prove the binomial~Poisson connection .
Note the table gives the probability <
x, just like a Normal table.
Read :Ash sec. 2-5.
HW: 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 Interactive Probability, Poisson
Process. Experiment: Number of arrivals in [0,t]
Do the Store of problem A, 2 hour period, 5 customers
per hour. (t=2, r=5) (Set Male prob. to 0) Run a few simulations.
Read off Distribution Table (right of graph) , blue Pr column, 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.
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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