Math300 Day 19

Math 300 , Spring 2008 Day 19, M, March 10 .. Hit reload ...

Midterm given out: Covers material through last HW, emphasis on probability problems and distributions. Joint distributions on X and Y (like red and black-sided chips). Rules for E(X+Y) but not for Var(X+Y).
Handouts; new algebra of (mostly) variances, 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, & we did it in class.
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 (b) 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.)

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

B) 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?
C) 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) )
Postpone: (Try now if you have time.) D) 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)

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)

= = = = = = = = = = = = = = = = = = = = = = = =
Algebra of expectation and variance (continued): (New Handout)
(Ash Ch. 7, pp. 220-235)
--Proved E (X+Y) = E(X) + E(Y).
Proved E(k) = k.   E(a+X) = a + E(X)

Go thru new 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?

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)

Old-- p. 228: Var k = 0, Var aX = a²VarX, Var (-X) = VarX, var (aX + b) = a²VarX
If X, Y independent: Var (X-Y) = VarX + VarY , Var (aX + bY) = a²VarX + b²VarY

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.... handout

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