Math 300 , Spring 2004, Day 27, F, April 9 Hit reload ..After 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  var(X) =

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  var(X) = 1/(lambda²)(calculate soon)

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 ). The average is  lambda per unit.
Since the length of the interval for W is x_o, the parameter for W is (lambda x_o)
            No successes in an interval of length x_o, Poisson prob. is exp(- lambda x_o) = P(X > x_o).
F(x) = 1- exp(- lambda x).
f(x) = F'(x) = lambda exp(- lambda x).
         (Sketch: f = 0 for negative x. f(0) =Lambda. downward curving from there)
               Gamma Applet  k=1, r = lambda. Run and watch the Poisson process happening.
                     (Vertical scale on left is in tenths, but goes above the max lambda.  Hard to read.)

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 (like the Geometric). (p. 124)

Another application:  Lifetime of something that has no "aging"--just subject to random "death".  lambda = 1/average lifetime.
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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.  Add them.   See p. 125 for the computation...
When we take the deriviative the sums "collapse" leaving the (still complicated) formula

 
 

(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.)

Gamma Applet , k is for waiting till k'th arrival.  In Quantile Applet, you can see the Gamma generalization: k is now the "shape" parameter, and can take on values between the integers.)

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Start Here Monday
Expected values and variance:  (back to ch. 7)
Instead of sums, we have integral signs.  E(X) = _-ooS^oo x f(x) dx
E(X) is still the balance point on the f(x) graph.
"Law of the unconscious statistician"--
If Y = g(X), then E(Y) = integral over all y's of y times density of Y = _-ooS^oo y f_Y(y) dy OR
                             = integral over all x's of g(x) times density of X = _-ooS^oo g(x) f_X(x) dx
   So E(X²) is the integral of x² times f(x).
E(a +bX) = a + bE(X) :  _-ooS^oo (a+bx) f(x) dx =_-ooS^oo (a f(x) + bx f(x)) dx = _-ooS^oo a f(x) dx + _-ooS^oo bx f(x) dx =
   a _-ooS^oo f(x) dx + b_-ooS^oo x f(x) dx = a + b E(X)

Var(X) =  E(X²) -µ² still.  (proof p. 225 still holds.)
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HW (read 4.3, including Gamma)  Ash p. 127  All Exponential until  #6c:
#1.
#2 (for e, you can use your Poisson table.)  #3e (cf. the bird calls sheet),  #4
#5
#6 "by time t" = "in t more minutes"
#7 [part f should say 4 particles--my book says 3.  My book also has 210 instead of 10 in the answer to g]
#8  Gamma

A.   Here's a picture of the F for the exponential, with a couple of percentiles marked.  Find a general formula for the p'th percentile by solving p = F(x) for x.  Check with the picture for p = .30, .75 (You'll need a calculator.)

B.  Substitute n = 1 in the Gamma formula for f, and check that you get the exponential (the exponential is the special case of the Gamma).

C. Review integration by parts.  Do this standard example:  integral from 0 to 1 of f(x) =  xe^x.

Hand in  D and E Wednesday (but you could do the reading now...)
Expected value:  Read pp. 213-16.
D.  Ash finds the mean (expected value) for the Uniform distribution on [a,b]  on pp 214-15. Understand that and check that the mean is the balance point on the graph (p. 103).
E.  Find E(X) for the first Density handout, f(x) = (x - .5), 1<x<2.  Check that it is above 1.5, as the balance point must be.