Math 300 , Spring 2008, Day38, W, April 30 Hit reload .After class corrected..

Mini-exam due Monday, Day 40. Continuous joint distributions through expectation.

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:  Finish HW on Day 36, Day 36.
 plus writeup of details of conditional prob's and E's for
f(x,y) = x+y, 0<x<1, 0<y<1.,  f(x,y) = 2e^-x-y , y>x, x>0, checking my work on Day 37 page
Also, fill in details for E(X|y), f(x,y) = 2e^-x-y , y>x, x>0.

Postpone 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.

= = = = = = = = = = = = = = = = = = = = = = =
New Handout:  Solution to A, Lognormal handout.

Continue looking at conditional prob's and  E(Y|x) Day 37.
 Questions on HW?

E(X|y), f(x,y) = 2e^-x-y , y>x, x>0.  Have f_X|y(x|y) =e^-x/(1-e^-y) , 0<x <y . 
       Look at DPGraph, Conditionals, 2exp(-x-y) on y gt x w margs.dpg
E(X|y) =   _oS^y  x e^-x/(1-e^-y) dx  We are integrating x's from 0 to y.
           = [1/(1-e^-y)]_oS^y x e^-xdx   
                Need to integrate by parts or use Ash's formula p.95: S x e^-xdx =  e^-x(-x -1)
                    When x = 0, e^-x(-x -1) = -1.   When x =y, e^-y(-y -1)
           = (1 - ye^-y -e^-y )/(1-e^-y)        Graph of E(X|y)

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Next time:
Continuing with Two random variables X, Y "jointly distributed".

IF E(Y|x) a straight line then  E(Y|x) = the "theoretical" regression line.  y-hat = E(Y|x)
Rho is the correlation coefficient, and the slope of the line if the standard deviations of x and y are equal.
Correlation coefficient = [covariance (X,Y)] / [(s.d. of X)(s.d. of Y)]  Ash p. 235
Recall for data:  the line goes through the (xbar, ybar) point.  Same here:  (µ_X, µ_Y) lies on the line.
The line formula yhat = a + bx  from xbar, ybar, s_x , s_y , r:
     Find b:   b = r  s_{y /} s_x
       Find a:  Solve  ybar = a + b xbar for a:  a = ybar - b xbar
HW:  Take the theoretical version above, solve for yhat, show that a and b follow the "same" rules.