Math 300 , Spring 2008, Day 26, W, April 2 Hit reload ..after class

HW   read rest of 4.2 Repeated from day 25
Ash p. 118
#11 f to F: Hint: move a bar across the page and watch how the area to the left grows, in each interval.
#3 P's from F (the little curved piece has formula 1/3 x²)
#7  (mixed)
#9 (F to f:  to check if it's a legal density you have to make sure F has no jumps)
A.  On Density-->CDF handout  ,  for x =1, 1.1, 1.2, 1.3, ....1.9,
  verify that  [F(x + 0.1) - F(x)]/0.1 =  f(x+.05) (f at the middle of the 0.1-wide interval)
B (added) Fundamental theorem of calculus:  http://cow.math.temple.edu/~cow
  Choose  Calculus Book II > 1.Integration > 4.Fundamental Theorem > 1.Differentiation and the Fundamental Theorem.  Do problems 1 thru 5.  (We would call all their f's F's  in these problems.)
You must click on 'Check your answer' to make the COW aware that you have entered an answer.   Help or Hints may be useful.
How to type the math ? (Mostly it works as you expect.  x^2 = x²):  Check your attempt with the Expression Interpreter. For rules  http://cow.math.temple.edu/~cow/Manuals/TypingHelp.html

HW Postpone this set
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 (read 4.3, including Gamma)  Ash p. 127  All Exponential here Postpone this set
#1.
#2 (for e, you can use your Poisson table.)  #3e (cf. the bird calls sheet),  #4
#5
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Exam 1 more solutions (yet another method for 5a)  Quiz returned. 
Comment on quiz:  1) "Mathematics is the reduction of the difficult to the routine."  2)  We work on different levels of abstraction--often the highest level of abstraction takes us away from the grubby/difficult.  But sometimes, in specifics, we have to go back down to the grubby.  Quiz: problem 1 was grubby calculation of an E, problem 3 was abstract manipulation of E's.
 Questions on HW?
Continuing with F(x) = P(X<x):  Day 25 for details
Properties: Day 25
Mixed distributions: Day 25
F to f (continuous)--take derivative!: Day 25

Start here Fri.
Applications of F(x) (Not in text).  The data version of the density function f is the histogram.
The data version of F(x) is called the "Empirical Distribution Dunction"--takes a step up at each individual observation (or is smoothed.)
   At each x, the proportion of individuals < x.
Survival curve S(x) = 1-F(x) = Probability (proportion) surviving past time x. (Goes from 1 down to 0)  Avalanches
   This site is very good on the statistics of cancer: http://cancerguide.org/scurve_basic.html

Using F(x) to find percentiles ("quantiles").  (Not in text)
  Def:   x_p is the p-th percentile  if 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(x) = p  is something we can solve for x ).
       Graphically, find p on the y-axis, go over to the F curve and down to x_p.    Quantile Applet, choose CDF, any distribution. (PDF = probability density function = density)

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) =q/p² 

Poisson:  A unit interval, with independent "arrivals" .   Y = # of arrivals in the unit interval. lambda = E(Y) = parameter.
P(n) = ?  P(0) =
  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).   Notice S(x) =P(X >x) just =  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)