Math 300 , Spring 2008, Day 27, F, April 4Hit reload ..after class.

Takehome mini-exam next week  on the continuous material, F, f etc, through 4.3 (exponential & Gamma)  & percentiles. 

HW (quantiles)
A.  On the  Density-->CDF handout  (solutions) ( f(x) = x-.5, 1<x<2) 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.)
B.  Use the  Quantile Applet, different distributions and parameters, to get used to these ideas. PDF = density.  Choose CDF, switch.  For the Standard Normal distribution, find the first and 3rd quartiles (25th and 75th percentiles.)  Also the four quintiles (20, 40, 60, 80%iles)--government economic data is mostly in quintiles.
C.  For (Ash p.106) the Uniform distribution on the interval [a,b], find the 25th percentile by graphical or geometric methods. Now find it by solving 1/4 = F(x) = (x-a)/(b-a) for x.  Check your answer.  Now solve p = F(x) = (x-a)/(b-a) for x, getting x_p, as a formula for the p'th percentile in a uniform distribution.

HW see what you can do--we found F(x) only.
Applets: Gamma, k =1 is Poisson. Gamma Applet r = lambda. Quantile Applet b = lambda.

(read 4.3, including Gamma)  Ash p. 127 Exponential (thru #5) +Gamma &Exp
#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. Yes Review integration by parts.  Do this standard example:  integral from 0 to 1 of f(x) =  xe^x. (Optional: review with http://cow.math.temple.edu/~cow
  Choose  Calculus Book II > 4.Methods of Integration > 1.Integration by Parts > 1.Integration by parts.) The COW is now linked from the bottom of the 300 index page.

B. Yes!   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 with e and ln.)

C. Postpone 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).

HW Hand in  A and B Wednesday, probably (but you could do the reading now...)
Expected value:  Read pp. 213-16.
A.  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).
B.  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.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  Questions on HW?

Continuing with F. Details Day 26.
--Applications of F(x) (Not in text).
--Using F(x) to find percentiles ("quantiles").  (Not in text)
--4-3: Exponential distribution Waiting time till first arrival in a Poisson phenomenon.
Monday, finish exponential, pick up here.- - - - - - -- - - - - - - - - - - - - - - -
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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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.)