Math 300 , Spring 2008, Day33, F, April 18 Hit reload to get most current versionafter class

HW:  Finish review of double integrals with COW as needed. With Ash 164-170.
Do Discrete problems repeated here from   Day 32:
HW Multivariate Discrete;  "MultDisc"  p. 3, all those exercises.  In 5.1.3c,f: There's not much of a double sum to write when the probability wanted is only one point.  What I'm looking for is that you try to write double sums that parallel what you do in writing double integrals, but in the sum case you can actually check your work by brute force.  Do what makes sense.
With them: p.4:  B) P (3 < X+Y < 4) for the two distributions on that page (those of examples 2.8.2 and 3)
Then A) Find P( 0 < X-Y<1) for f(x,y) = xy2/30 , x = 1,2,3,4, y = 1,2 (dist. of example 2.8.2 in handout)

HW Multivariate Continuous:  
A) Write up completely the work we did in class with g(x,y) = y,  0<x<4, 0<y<3;
    Finding C =18 to create density f(x,y) = g(x,y)/18, finding P(X+Y < 4), finding marginals.
Do the above for Mon: Also start the ones below, graphing the regions (bolded parts)
DO on Handout  "MultCont", #17 and #18 (p. 2): First, Sketch the region of positive density of f, and do parts a.
    Next: Also determine their marginal densities and whether X and Y are indep.for 17,18,19, and finish 17, do 18b, 19b.
    17, 18, 19 are in DPGraph files to be viewed.(the z axis is distorted to get a better picture.)
Ash p. 181, #2a For this assignment, graph the region of f, and the region you want the prob. of. Next, finish it.  (Hint--p. 164 has pictures for min and max--but in my book the text refers to the wrong pictures (the captions are right).
Ash p. 181,  #7.  Do it by graphing the regions, and since all pairs x,y are equally likely, find the prob's by proportional areas (cf. Ash pp. 31-2, and #7 p. 35.)

= = = = = = = = = = = = = = = = = = = = = = = = 
Last questions, transformations of random variables?

Two random variables X, Y "jointly distributed".  Ash Ch. 5.  First 5.1 and 5.2
In practice: Two random variables X, Y  measured on the same experiment.
Sample space:  points in x-y space.
   Probability of a region:  sum or integrate over the region.  Discrete  Day 33

"Marginal" probability/density function:
      Function for just X (not "looking at" y):   pX(x) or fX(x)
                    or just Y (not "looking at" y):   pY(x) or fY(x) .
X and Y INDEPENDENT:  p(x, y) =  pX(x)pY(y),  f(x,y) = fX(x) fY(x) for every pair (x,y)

Continuous (p. 171 ff + "MultCont") Joint density function f(x, y)--a surface above the base x-y space.
    Probability of a region R  in x-y space= volume under f(x,y) and above the region.
The total probability (volume) has to = 1.

Integrals:   First do total volume, then do P(X+Y < 4)

In the inner integral, the "other" variable acts like a constant.  Try it with g(x,y) = y,  0<x<4, 0<y<3

Find Volume  under the curve: Dividing g by this will give a function with volume 1, a density..
   Let f(x,y) = Cg(x,y) = Cy as above.C = 1/18
 Now  Find P( X+Y<4) by integration.  = (1/18) 0S3 (yx|0x=4-y) dy = (1/18) 0S3 (4y -y2) dy = (1/18)[4·32/2 -33/3] = 1/2

    Find the marginal distributions:

get f(x) = (1/18)(9/2)=1/4, 0<x<4
fY(y) = (1/18)y·4 = 2y/9, 0<y<3
Are X and Y independent?  Check.
NOTE:  X and Y canNOT be independent unless their joint support (region where p or f is Not 0) is rectangular!
Because if support isn't a rectangle, there will be a pair (x,y) where X and Y have positive probabilities/densities, but the pair has 0 prob/density.

3-dimensional graphs of HW problems: Download DPGraph from source.  We have a site licence--go to Wells College and download.  In Macmillan 110, on desktops of some machines, Class Material Folder/ Math 300/ DPGraph08/.  Open DPGraph Read Me 08 file, read what to do.  It should run ok from in there; if not, install it and get it from the Program menu. Files for HW in For 300 Class folder (Clicking on a data file won't find the program unless it's in the Program menu, but you can open any data file from within the running program.) Backup folder contains copy of everything.   You can copy the whole DPGraph08 folder to your own machine, it should run fine.

DPGraph images.  Double click?  Or start DPGraph and open from within.
   Turn image around with arrow keys, shrink/stretch with PageDown/ PageUp.  Scrollbar menu item, slice x, y, or z, then use right scrollbar to move slice.  Show x(1+y)andx+y=1.dpg , which I used for the images above.

Sievers home
 Math300-Sp08/Day8p33.htm 
2pm
 4/20/08

This page belongs to Sally Sievers who is solely responsible for its content. Please see our statement of responsibility.