Math 300 , Spring 2002, Day 18, F, March 8 Hit reload to get most current versionAfter Class

Expectation:
mean of geometric, using formulas on p. 36, Ash
 
Handout, p1 arrows-->. If W is a function of X (W=g(X)), you can find E(W) either by
  >> summing g(x_i)·P(X=x_i)   for all the values of x_i,    or by
   gathering all the probabilities of  the x's which have the same w-value, thus finding the distribution of W, and            ·
 >>  summing w_j·P(W=w_j)   for all the values of w_j.
  Usually we use the first way, as when we find Var(X) = E(X - E(X))² , but both work.
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Def: X and Y are independent random variables if
       P(X = x and Y = y) = P(X = x) ·P(Y = y) for  every possible  pair x and y.
  What happens on X has no effect on the probabilities of Y:  P(Y = y) = P(Y = y | X = x)

If X and Y are independent, then E(X · Y) =  E(X) · E(Y)  (will prove, next week.)  This is needed to prove
         Var(X+Y) = Var(X) + Var(Y).
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Finding E's in complex cases:  If X = X₁+ X₂+...+X_n then E(X) = E(X₁)+E(X₂)+...+E(X_n). (Ash 3.2)
        (If the X_i's  are independent (not usually true), then the variance can be found the same way.)
         If the X_i's take on the values 0 or 1, they are called indicator random variables, and E(X_i) = P(X_i = 1)
    Binomial, drawing without replacement (mean only): X_i =1 if i'th trial is a Success.
Related idea: p. 77 #5:  If the sample space is {0, 1, 2, 3, 4,...} (or {1, 2, 3, 4,...})
   E(X) =  sum_{i=1 to infinity}  (P(X > i))  =
                                             p₁ +  p₂ +  p₃ +......
                                                  +  p₂ +  p₃ +.....
                                                           +  p₃ +......
                                           1p₁ +2 p₂ + 3p₃ +....      = sum _{i = 0 to infinity}( i·p_i)
       Reread Moore pp.380 for derivation of Binomial mean and standard deviation.
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If time: proof of E (X+Y) = E(X) + E(Y),
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Read Ash Sec 3.2, 3.4.  Skip 3.3.   Next, Ash Ch. 7, pp. 220-235 (ignore anything with integral signs)
HW:
Prove E(kX) = kE(X).
From Ash (These are all tricks of one sort or another.  Don't be discouraged if you can't do most of them without looking.)
Use p. 77 #5 to find E(X) for a geometric distribution (another way)
   p. 84, 1 + See  addition on Day 17 handout  p. 4
      3, 4, 5,
      6 optional
  p. 92, 7, 10, 17.
     Others Optional, but good: see the list on the Day 17 handout p. 4.