Math 300 , Spring 2004, Day40, M, May 10Hit reload... After class

E(Y|x) a straight line--> 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.

http://www.math.uah.edu/stat/applets/BivariateUniformExperiment.xml
Try with the shape = triangle.  Regression line from data is in red, "theoretical" is in blue (compare with handout triangle)  Run 100 points to see shape of support of distribution.
http://www.math.uah.edu/stat/applets/BivariateNormalExperiment.xml
Rho is the correlation coefficient, and the slope of the line if the standard deviations of x and y are equal.  See handout on bivariate normal.
 - - - - - - - - - - - -
 We know how to find the mean and the variance of W = X+Y.
--Last semester you were told that the sum of two Normal random variables is normal.
 --It seems reasonable that the sum of two Binomial random variables X B(n,p) and Y B(m,p), with the same p,  should be X+Y Binomial B(n+m, p), since the first models flipping a coin n times and the second models flipping the coin m times.  Doing them in sequence should be the same as flipping the coin m+n time.
--Likewise it seems reasonable that the sum of two Poisson random variables X+Y should be Poisson.  Suppose that X counts the number of arrivals over a time length lambda _x and that Y counts the number of arrivals over a time length lambda_y  (then for each the expected number per unit interval is just 1.  The sum X+Y will count the number of arrivals over a time length (lambda_x + lambda_y) .  (Ash gives a different, also good, rationale, p. 202)

 Question:  How would you find the distribution of W = X+Y, in general?
Try it on discrete:  see handout.
             f(x,y) = (x+y)/21, x = 1,2,3, y= 1,2.  (recall sum of 2 dice)
w:       2 ,         3 ,                 4 ,              5
 f:   (1+1), (1+2)+(2+1), (2+2)+(3+1), (3+2), all over 21

Answer:  For f(w) for a fixed w, Add on lines x+y=w, or over x, with y = w-x. i.e.
  For each w_o, Sum f(x, w_o-x)over all the x's for which f(x,w_o-x) is positive. 
  (Ch. 6.1 does more of the discrete.)
  Continuous:  Use  method of CDF, transforming random variables, find that we can get f(x,y) by
      integrating on lines x+y = w, or x = w-y.   Details on handout.   (Ash gives a dx argument)
If X and Y are independent, the resulting integral is called the Convolution of the two marginals: Important enough for its own notation, f*g.  These occur all over mathematics, especially when dealing with linear models.

DPGraph shows (Last Things folder) slices of the areas that get integrated in this process, for  our familiar examples f(x,y) = x+y on unit square,  f(x,y) = 2e^-x-y  on the triangular region.
Another new topic:  Bivariate normal distribution: (handout)

  Correlation coefficient rho = cov(X,Y)/stdev(X)stdev(Y)  (cf. sample correl. coeff.)
    This is also a parameter in the joint normal.  Not coincidentally, it turns out to be the correlation coefficient for X,Y.
     Level curves are ellipses.
      E(Y|x) is line that bisects "vertical" distances inside ellipses.
DPGraph pictures (Last things folder).

--- --- --- --- --- --- --- --- ---
HW  Read 6.1, with as much of the computational detail as you can stand; at least the "intuitive" examples.
A.  From the handout, carefully graph the final f(w) = w², 0<w<1; = w(2-w), 1<w<2.
B.  Find f(w) for W = X+Y for f(x,y) = 2e^-x-y  , 0<x<y.  Integrate with respect to x.  The only tricky thing is figuring out what the upper limit on the dx  interval is:  it is where the boundary line y=x intersects the line w=x+y.  For a fixed w, (say w=3) eliminate y, finding the upper limit of the x-interval as a number.  Repeat for a fixed w_o.  You should get a recognizable distribution for f(w).
p. 207, Read all problems, guess answers, check with back of book (good review of many distributions.)

Read Normal handout, as much as you can stand.
This is the last assigned HW.  I'll elaborate on this, answer questions, and show you why 1/sqrt(2pi) is the right thing to make the normal curve have area 1--Wed and Fri.

Final exam:  Takehome, comprehensive, available Friday, due on or before Thursday May 20 at 4pm.  I'll be on campus part of Monday, Tuesday morning, 9:30-4 Thursday.  If you want/need to hand in your exam after I leave Thursday, you must deliver it to me at my home in Ithaca, not later than Saturday noon.