Math 300 , Spring 2002, Day 20, W, March 13 Hit reload to get most current versionWed. Night

Algebra of expectation and variance (continued):  (see Handout)
Correction: in proof of (5), right hand column: Let y_jP(Y=y_j) = b_j, not y_j = P(Y=y_j) = b_j
(Ash Ch. 7, pp. 220-235)
--Proof of E (X+Y) = E(X) + E(Y), handout.
--#E, Day 19:  E(X(X-1)) = E(X²-X) =E(X²)-E(X)
--E(X(X-1)) for Geometric dist.

VarX = E(X²) - (E(X))²     (We can turn this around and find E(X²) from VarX and E(X))

If X and Y are independent, then E(X · Y) =  E(X) · E(Y)

 Monday after break:
 Cov(X, Y) = E[(X-E(X)) · (Y-E(Y))] (def.)    = E(X · Y) - E(X) · E(Y)
    Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y)    (a cov term for every pair, if summing more than 2)
          If X and Y are independent, Cov(X,Y) = 0, and we get our familiar Var sum.
    Correlation "rho"of X, Y is Cov "standardized" by dividing by both standard deviations (p. 235).
            is Theoretical version of correlation coefficient r.

I'll go over variance of Hypergeometric, with handout.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
(Re)Read Ash pp. 220-235 (ignore anything with integral signs)
HW:   from list on  bottom of Algebra handout; repeated here.
Due Monday after break:  these + the last day's list.  (except for the yellow ones, #8, 13,14)
p. 233  #10
A. Show how to find Var(X) if you know E(X(X-1)) and E(X). (Hint: use (3),p.225, the results from problem E, Day19, and simple algebra.)
B.  Use your result from Day 19, #D, and #A just above, to find the variance of the Poisson.

#8,13,14 with Monday's
#8 (Note that if E(XY) = E(X)E(Y),  then. Cov(X,Y) = 0.  Use this & your results from p.223#14)
#13, #14 (finding Covariances by brute force)

#19a Var for geometric = q/p² Part will be done in class, + #A
# 19b Var for neg. binomial.  Use the fact that Var of geometric is q/p², and the "trick" of p. 82 of looking at the neg. binomial as the sum of k independent geometric random variables.  (You may do it for k = 3, as p. 82 does.)

Continue to fill out your Named distribution sheet.  You've now proved or should have read the proofs for:
   Binomial (Mean and Variance)
   Geometric (Mean: 3 ways--2 in class, 1 on p. 81) #19a above
   Hypergeometric (Mean, p. 80--same as Binomial.  Checked in an example)
   Poisson (Mean: in class, also p. 76)  #B above