Math 300 , Spring 2004, Day 17, W, March 10Hit reload ...

  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.

Prove: Prove E(kX) = kE(X). 
         (Use the above to justify finding E(kX) using dist. of X.)   Assume P(X = x_i) = p_i for notational ease.
Proof:  E(kX) = Sum_i[(kx_i) p_i] = Sum_i[k(x_i p_i)] = kSum_i[(x_i p_i)] (pulling k out of sum: distributive law)
           = kE(X) by def. of E(X).
- - - - - - - - - - - - - - - - - - -
Need to do:  proof of E (X+Y) = E(X) + E(Y) (handout, will go over in class)
- - - - - - - - - - - - - - - -
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), any x,y.

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).
- -  - - - - - - - - - - - - - - -
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, n drawings without replacement (mean only): X_i =1 if i'th trial is a Success. E(X_i) = p
                E(X) = Sum[ E(X_i)] = np
                Var(X) = Sum[Var(X_i)]
                     (remaining: variance of one Bernoulli trial S--was HW)  E(S)= p
                     Var(S) = E(S - E(S))² = (0-p)²P(S=0)+ (1-p)²P(S=1) = p²q + q²p = pq(p+q) = pq
              Var(X) = npq
  (Re)read Moore pp.371-3 for derivation of Binomial mean and standard deviation.

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)

Proof of E (X+Y) = E(X) + E(Y)
 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Read Ash Sec 3.2, 3.4.  Skip 3.3.   Next, Ash Ch. 7, pp. 220-235 (Variance. Ignore anything with integral signs)
HW:  A. Add to your "known distributions" page the means found in sec's 3.1 and 3.2.
B. a. Prove E(kX) = kE(X) as above, only when n = 3.  Write it out with +'s.
                Start E(kX) = (kx₁)p₁+(kx₂)p₂+(kx₃)p₃ =...
        b.  Prove E (a+X) = a+E(X), first with n = 3 and +'s, then in general with i's and Sum (big Sigma) notation.
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.)
C. Use p. 77 #5 to find E(X) for a geometric distribution (another way) 
     Hint:   p₃ + p₄ + p₅ +...=P(X>2); you have a simple formula for that.
   p. 84, 1 + See  addition on Alg of exp. handout  p. 4
      3, 4, 5,
      6 optional
  p. 92, 7, 10, 17.
     Others Optional, but good: see the list on the Alg. of exp.  handout p. 4.

D)Derivatives review: Find the derivative with respect to w:
  (w³- h)/(3+w + 2w²),    exp(-w²),        ln(2 + 3w² )
   Find the second derivative with respect to q:    q,  q², q³, q⁴, q⁵, q⁶,   q^x

E) Graph the four points in the x,y plane (0,2), (1,0), (1,4), (2,2).  Let each point be equally likely (prob = 1/4).  Let X be the value on the x-coordinate, and Y be the value on the Y coordinate.
  a) Find E(X + Y) by evaluating x+y for each of the 4 points, multiplying by the probability, and summing.
  b) Find E(XY) by evaluating xy for each of the 4 points, multiplying by the probability, and summing.
  c) Find the probability distribution of X.  Find the probability distribution of Y.  Find E(X).  Find E(Y).
  d) Check if E (X+Y) = E(X) + E(Y).
  e)  Check if E(X · Y) =  E(X) · E(Y)
  f)  Check if X and Y are independent.  (For independence, for any x,y pair, P(x,y) = P(X=x) · P(Y=y).  So to show non-independence, you only need to find one x,y for which that is not true.)

F)  Another Data problem:  Handout on Bird Calls.  Do all the questions (including 1.6.4, frizzle fowl).  Recopy table 1 to have enough room to fill it in.  Est. rel. freq. is just the probabilities.