Math 300 , Spring 2004, Day 23, W, March 30 After class Hit reloadCorrected 4/2

Quiz 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)

Logic in the trenches:
( 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"
To show  ( If Athen 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. ( If "Cov(X,Y)Not 0", then "X,Y Not independent".)

(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 measure, not a count.

   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.

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 (Old handout, or get new)  --shows the step function
 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

CDF from probability function:
                                    F(x) = P(X < x) step function  (graphed in class)
x    1   3   4   6                                    0,        x < 1
p   .2  .2  .3  .3                                   .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.
(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)

HW:
Some more continuous probability problems: Handout "Density problems"
  Do #1, 3b, 4b, 5

Read 4.2 pp.104 to 114.
Discrete CDF's:
A.  Poisson Distribution: Use the Poisson table handout to find, for lambda = 2.2,
    P(X< 4), P(X > 4),  P(2 < X < 4),  P(2 < X < 4)
B.  Use M&M's binomial table, and construct a table column for B(4, .5), which tables F(x). Check with the cumulative  binomial table handed out (Back of Poisson table).  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)

Continuous CDF: graphing
A. Normal Distribution: Tables often give cumulative distributions; almost always for Normal.  (There is a table on p. 129, Ash--"Unit normal", or use the one in M&M).
a) (Review.) Use  P(a < X < b) = F(b) -F(a) and a Normal table to find P(.5 < X < 1.5) for X the Standard normal distribution.
b) Use the table for the Standard Normal Distribution and plot P(X < x) vs. x, for x = -3, -2.5, -2, -1.5, etc. by .5's, to +3. Connect the dots with a smooth line, and you will have graphed the CDF of the Standard Normal. (There is a table on p. 129, Ash--"Unit normal", or use the one in M&M)
Postpone:
B.  Density-->CDF handout: Graph CDF by counting squares, carefully.