Math 300 , Spring 2002, Day 29, W, April 10 Hit reload to get most current versionAfter class

4-3: Exponential distribution
Binomial: n trials, probability p of success on a single trial.
   Waiting time X till first success:  Geometric distribution.
         P(X= 6) = qqqqqp, 5 trials with NO successes, before the success.      E(X) = 1/p

Poisson:  A unit interval, with independent "arrivals" .   Y = # of arrivals in the unit interval. lambda = E(Y) = parameter.
    Waiting time X till first success:  Exponential distribution.  E(X) = 1/lambda  (calculate later)  (I didn't say this in class)

Finding distribution of Exponential: Find F(x).
F(x_o) = P(X <  x _o) = 1 - P(X > x_o).
   P(X > x_o) is the probability that in the interval (0,   x_o), there are NO arrivals.
    Find that probability by using the Poisson distribution .
          Let W = # of arrivals in the interval (0,  x_o ), average lambda per unit.   The parameter for W is lambda x _o
            No successes in an interval of length x_o, is exp(- lambda x _o) = P(X > x_o).
F(x) = 1- exp(- lambda x).
f(x) = F'(x) = lambda exp(- lambda x).   (sketch)

Like the Geometric, it's memoryless:  Start anywhere, it's as if there was no past. (Because in the Poisson, arrivals are equally likely everywhere)
If you have waited for time t_o already, the probability of having to wait  at least x more is the same as just having to wait at least x, starting from 0:
 P( X > x+t_o |X > t_o) = P(X > x).  Proof, done.

Another application:  Lifetime of something that has no "aging"--just subject to random "death".  lambda = 1/average lifetime.
  (I didn't say this in class)
HW (read 4.3, including Gamma)  Ash p. 127
#1.
#2 (for e, you can use your Poisson table.)  #3e (cf. the bird calls sheet),  #4
#5
#6
#7 [part f should say 4 particles--my book says 3.  My book also has 210 instead of 10 in the answer to g]
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Start here Friday
Gamma distribution with parameter n:  analogue of Negative Binomial:  Waiting time till n'th arrival.
  Strategy for finding distribution.  Find F(x_o) = P(X <  x _o) = 1 - P(X > x_o).
   P(X > x_o) is the probability that in the interval (0,   x_o), there are NO arrivals, Or 1, Or 2,.....Or n-1. (the n'th is yet to come.)
      Each of those is a Poisson probability.   See p. 125 for the computation.
(Why "Gamma"?  The gamma function is a generalization of the factorial k! to values of k that are not integers.  The Gamma distribution has a factorial in the denominator.  It can be generalized to use non-integer n as parameter (but doesn't have the interpretation of waiting till n'th arrival any more.))
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Using F(x) to find percentiles.
  Let x_p be the p-th percentile, that is P(X < x_p ) = p
 For example, the median x_0.5  is the number such that P(X < x_0.5 ) = 0.5, the number with probability 1/2 below it.
  Since F(x) = P(X < x),  F(x_p ) = P(X < x_p ) = p
So to find x_p , we can set F(x) = p and solve for x (assuming F is something we can solve for x in).
       Graphically, find p on the y-axis, go over to the F curve and down to x_p.
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HW  Ash p. 127
#8  gamma
A.  On the Density sheet (with the graphs) use the graphical method (over and down) to find, approximately, the median.  The 80th percentile.  Then solve the equation  F(x_p ) =  p  for the x's for p = .5 and .8.  (Check the two methods give approximately the same answers.)