Math 300 , Spring 2008, Day 22, M, March 24 Hit reload.after class.

Activism day  http://aurora.wells.edu/%7Esymposium/keynotes.html Friday--hold class? Email me.
Quiz Friday ?Monday?: Closed book, Expected value stuff, chosen from:

   Deriving, E(X(X-1))of Poisson. (for sure)
   Var(X)  from E(X(X-1)) and E(X)  (how to find & why)
   Proofs for Var and Cov alternate formulas.  Var(X+Y) = Var(X) + Var(Y) +2Cov(X,Y) (derivation )
   True/false: X,Y Independent <=?=>  E(XY)=0, Cov(X,Y)=0  (like Ash HW)
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HW:  Read 4.1. Postpone most, but you can go through and graph all the densities, leaving space for the computations.
Note:  f(x) = |x| is "already" a piecewise function:  f(x) = x if x > 0,  = -x if x < 0  See p. 96 for examples of use.
p. 103-4, #1 thru 6
Read ahead pp.104 to 114.  (You can do the following problem whether or not the reading makes sense.)
do A.  Wrong graphs?  p. 96, graph fig. 2 is wrong in my text.  It shows a smooth curve, with the slope = 0 at x = 0.  Check yours, regraph and fix it if it's wrong.
Likewise p114 graph fig. 16 is wrong in my text. Graph it, and see if yours is right or wrong.  (The formula is just above the graph.)
Some more continuous probability problems: Handout "Density problems"
  Do #1, 3b, 4b, 5  Hand in!

Read 4.2 pp.104 to 114.  Discrete to start with.  Then continuous, then mixed.
Postpone Discrete CDF's:
A.  Poisson Distribution: Use the Poisson table handout to find, for lambda = 2.2,
    P(X< 4), P(X > 4), P(X = 2),  P(2 < X < 4),  P(2 < X < 4)
B.  Use M&M's binomial table (Table C) (or the Excel version), and construct a table column for B(4, .5) which tables F(x). Check with the Cumulative  binomial table here.  (Many books give cumulative tables.)  Also Graph it.
   Find P(1 < X < 4) from M&M's table, and from the cumulative table.  Show your work.

Ash p. 118, #1  Write the formula for F(x) carefully, paying attention to endpoints.(do graph both graphs)

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
A little Logic in the trenches: 
    True/false: X,Y Independent <=?=>  E(XY)=0, Cov(X,Y)=0
( If A then B)   is logically equivalent to   (If Not-B then Not-A) ("contrapositive")
In mathematics, we say "If A then B" usually meaning "in all possible cases, If A then B"
   (If "X,Y independent", then "Cov(X,Y) = 0"). Always True. ( If "Cov(X,Y)Not 0", then "X,Y Not independent".)
To show  ( If A then B) is false, you must demonstrate a case where A is true and B is false.
     (a "counterexample").  That is, a case where A is true, and Not-B is true.
        (or looking at the contrapositive, where Not-B is true and Not-A is false,
                    that is where Not-B is true and Not-Not-A is true.
        (If "X,Y independent", then "Cov(X,Y) = 0"). Always True. We can't find a counterexample, where "X,Y  independent"  and "Cov(X,Y)Not 0")

(If A then B) true does NOT say anything about (If B then A)  (the "converse")
     It can be that A is false and B is true, which would make (If A then B) "true" (it doesn't contradict it),
             but makes (If B then A) false.
               A HW had "X, Y  Not independent", but "Cov(X,Y)= 0".  (counterexample to (If B then A))

Ch. 4, Continuous distributions--outcome is a (continuous) measure, not a count.
E.g. Weight of pumpkin from a certain variety, wait time for first cosmic ray to hit Geiger counter, spinner...

Questions?  f(x) on p. 95 and g(x) on p. 96 --piecewise-defined functions

Physics: Density (of a rod...) = amount of mass per unit interval. If density varies along the length of the rod, we can talk of the density f(x) at a point x.  The mass in a short interval of length dx at that point is (density x interval-length) = f(x)dx.   Mass between a and b is sum of masses in short intervals:  very short intervals, and you have _aS^b f(x) dx.
Let mass be our metaphor for probability, area our representation for mass=probability.

Density curve:  f(x) > 0, with total area under f(x) = 1.  "Probability density function" = "pdf"
Probability = Area under the curve.  P(a<X<b) = area between a and b. = _aS^b f(x) dx. 
    1= _-ooS^oo f(x) dx

We'll use integration to find areas under the curve (usually.  The Normal density can't be integrated with a formula).

--The total area under the curve being 1 forces this:  as x goes toward plus, or minus, infinity, f(x) goes to (or is) 0.
--  _aS^a f(x) dx = 0,  so P(X = a) = 0 for any continuous distribution.

4.1:  Newish from calculus: Piecewise defined functions.  You have to make sure you're integrating the right formula; break the integral at points where the formula changes.

Start here Wed: Any function f(x) > 0 can be made into a density by dividing by a number that makes the result have total area 1.
   What number to divide by?  The area under f(x).

Example:
 f(x) =  x-2,  2< x < 4                     Graph it.
        = (x-4)²,  4< x < 5                Is it a density function?
        = 0 elsewhere.                        1 ?=?  _-ooS^oo f(x) dx
_-ooS^oo f(x) dx =  0 + ₂S⁴ (x-2) dx  + ₄S⁵(x-4)² dx + 0 =....=....= 2 + 1/3 = 7/3.
f(x) is not a density function.
But g(x) =(3/7) f(x) is!  Call the r.v. with this density function X.
  P(X< 1) = 0
  P(X< 3) =  _-ooS³ g(x) dx = ₂S³ (3/7)(x-2)dx  = ....= 3/14
  P(X > 3) = 1- P(X< 3) =  11/14
             or =  ₃S^oo g(x) dx = ₃S⁵ g(x) dx =  ₃S⁴(3/7)(x-2)dx +  ₄S⁵(3/7)(x-4)²dx

I won't try to put all  the calculus computations on the webpages--too time-consuming.  So take good notes, check with one another for "repairs" to notes.

4.2:  Cumulative Distribution Function (CDF, "Distribution Function")
F(x) = P(X < x).    Defined for every x on the real line.  Capital letter. (pdf uses small letter)
Continuous: Area to the left of x under the density.  Continuous function.
  P(a < X < b) = F(b) - F(a)      (You used this to find Normal probabilities in Statistics)

Discrete: Sum of probabilities to the left of x (including x).  Jumps at each lump. "Step function."
   See Poisson table   --shows the step function.  Note at jump, how -(  shows the F(x) value not there.
 P (X < a) = F(a), P (X < b) = F(b), so
P (a < X < b) = F(b) - F(a)  (note missing = at left end)  Must watch ends carefully
  For lambda = 4:  P(X< 3) = F(3) = .433        ---0---1---2---3---4---5---6---7---
            P(X > 3) = 1 - F(3) = .567 = P(X > 4)
         P(4 < X < 6) = P(3 < X < 6) = F(6) - F(3) = .889-.433
       P(X=3) =   P(X< 3) - P(X< 2) = F(3) - F(2)  = .433  

CDF from probability function:
f(x) or p(x)  (Ash uses p(x))      F(x) = P(X < x) step function  (graph in class. Show endpoints, like Fig. 3.) 
x    1   3   4   6                                  ---0---1---2---3---4---5---6---7---
p   .2  .2  .3  .3                               0,        x < 1
                                                     .2,   1< x < 3
                                                     .4,   3< x < 4
                                                     .7,   4< x < 6
                                                     1,    6< x
Continuous: Area to the left of x under the density.  Continuous function.  Details to follow.
(Some) Properties:  F(x) --> 1 as x --> infinity (it may actually get to 1, earlier)
                               F(x) --> 0 as x --> minus infinity (as x decreases)  (it often gets to 0)
                  F(x) is NONDECREASING (it may be flat or increasing as x increases, but it's never decreasing)