Math 300 , Spring 2008, Day 39, F, May 2Hit reload ...

Mini-exam due Monday, Day 40.
DPGraph08 folder is now in Class Material on all computers in Mac 101 except the 3 closest to the lake. NOT in Mac 110.

HW: Regression:  Solve this equation

 for yhat.  Show that the slope b and the intercept a
(when the line is in the standard form yhat = a + bx) follow the "same" rules as for data--
   b = r  s_{y /} s_x    and   a = ybar - b xbar.

= = = = = = = = = = = = = = = = = = = = = = = = = = =

Continuing with Two random variables X, Y "jointly distributed".
f(x,y) = 2e^-x-y , y>x, x>0.  Have f_X|y(x|y) =e^-x/(1-e^-y) , 0<x <y .   An exponential distribution truncated at the right end.
Graph of E(X|y)   Movement to right as y increases appears to "slow down."
E(X|y) =   (1 - ye^-y -e^-y )/(1-e^-y)   = 1 - [ ye^-y /(1-e^-y)].   As y gets large, the [  ] term --> 0.  (Denominator -->1, top --> 0)
   So as y gets large, E(X|y) approaches the line x = 1 asymptotically. 
   This makes sense:  The expected value of the exponential distribution e^-x is 1; as the point of truncation moves farther right, the conditional distribution of X given y looks more and more like e^-x .

E(Y|x) : the "average" y given a particular x. IF it's linear, it coincides with the (abstraction of) the usual least squares best fit line. See Day 38 for details.

Bivariate Normal distribution: ("Joint Normal"). Parameters: 2 means, 2 s.d.'s, and rho (correlation coefficient)  Handout
   Level lines are concentric ellipses. See Day 38 for details.
Forgot to mention there, that the conditional distributions are also normal:
  and E(Y|x) is the usual regression line (in the standardized situation, E(Y|x) = rho).

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Functions of 2 random variables (Ash. 5.3); especially W = X+Y (Ash Ch. 6.1) a brief overview.
   We know the expected value E(W) = E(X) + E(Y), and,
      if X and Y are independent, we know Var(W) = Var(X) + Var(Y)
But we did that without knowing what the distribution of W was!

For some distributions, we can tell from their interpretation what the sum should be (Ash pp. 201-3)
Binomial: If X₁ is B(n₁, p) and X is B(n₂, p) (same p!) and they are independent, then W = X₁ + X₂ is B(n₁ + n₂, p)
   Why? Do the (same) Bernoulli trial n₁ times, then another n₂ times. # of successes is W.
    Ash gives similar arguments for Poisson, Exponential-->Gamma.  (Read the non-computational "proofs")

Normal:  You were told in Math 251 that if you sum two Normal random variables, the result is Normal. 

On Day 1 you found the probability function for the sum of 2 6-sided dice.
  You summed along the line x + y = w, to find p(w). 
  What to do in continuous context?  Handout X + Y
              Find  F_W(w_o) = P(W < w_o) = P(X+Y < w_o), area below and to left of x + y = w_o line.
                 To get f(w), take derivative of F.
Try it by geometry on Uniform distribution on square:  f(x, y) = 1, 0< x, y < 1.