Math 300 , Spring 2004, Day 19, M, March 15 After class Hit reload ...

Go thru handout, as we work through more results of Algebra of expectations.
 If X and Y are independent, then E(X · Y) =  E(X) · E(Y) , I'll prove it.  Today??Soon...

New this semester:
   VarX = E(X²) - (E(X))²= E(X²) - (µ_x)²   (p. 225) We can turn this around and find E(X²) from VarX and E(X))

   Cov(X, Y) = E[(X-E(X)) · (Y-E(Y))] =  E[(X- µ_x) · (Y- µ_y)](def.) (HW) = 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.
           Cov(X,Y) = rho · sigma_x · sigma_y   (cf. M&M  rule 3, p.330)

Why these "algebra" rules?  To develop ways to simplify or clarify finding of means, variances, covariances; use abstraction rather than the brute force of computing down and dirty in  double sums etc.
I'll go over variance of Hypergeometric....
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Read Ash pp. 220-235 (ignore anything with integral signs)  Especially note proof of "More practical way to find VarX, p. 225."  It should be understandable.
HW:  Many are taken from list on  bottom of Algebra handout; repeated here.  Eventually you'll do them all.
p. 223, #14.  (If we haven't yet proved that "If X and Y are independent, then E(X · Y) =  E(X) · E(Y)",  we've talked about it, and it's true.  Your results from the problem on the points (0,2), (1,0), (1,4), (2,2) may be helpful here also.)

p. 233,  Accept the alternate formula for variance: VarX = E(X²) - (E(X))²  (formula and proof, p. 225) and use it where useful.
    #1, #3
    #6 a.  We'll work toward proving it next time.
    # 7,  using the rules you learned from Moore & McCabe, & VarX = E(X²) - (E(X))²
    #16 (they mean: you know the distribution of X if you think about it.)

Postpone:  (Try now if you have time.) A) On a separate page, find E(X(X-1)) for the Poisson distribution.  Keep it for next time.  (Make a table: the beginning is shown on the bottom of the Algebra handout.  Then you can use the trick on p. 76)

B)  Use the algebra of Expected values to get a relationship among  E(X(X-1)), E(X²), and E(X).  Hint: X(X-1) =??

C) Algebra refresher:  You know the expansion of (a+b)²= a² + 2ab+ b², I trust.  One cross product, 2ab.  Expand (a+b+c)².  How many cross products and what are they?  How many cross products will there be if you expand (a+b+c+d)²?
D) New:  Use the algebra of Expected values to show that E[(X- µ_x) · (Y- µ_y)]
      = E(X · Y) - µ_x · µ_y   (which = E(X · Y) - E(X) · E(Y) )

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)
   Hypergeometric (Mean, p. 80--same as Binomial.  Checked in an example)
   Poisson (Mean: in class, also p. 76)