Math 300 , Spring 2002, Day37, M, April 29 Hit reload to get most current versionAfter class

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

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In class:  Worked on  Probability Handout problem #17:  Density f(x,y) = Cx(1-y), 0<x<1, 0<y<2.
   Found C = 1/2.  Found f_X(x) = 2x, 0<x<1.   f_Y(x) = (1+y)/4, 0<y<2.
  Now we can see that X and Y are independent.

Back to Expected value (Ash ch. 7):  (law of the unconscious statistician again:)
 You can find E(X) just from f(x), or from f(x,y).  You can find E(XY) only from f(x,y) (unless X and Y are indep.)
  You can find E(X+Y) from f(x,y), or by finding E(X) +E(Y).  These things work because of the rules of iterated integration, where the variable not being integrated acts like a constant for the moment.  Reread Ch. 7-1, the parts with integration.

We began finding E(XY) for the density above.  You are integrating xyf(x,y), which = (x²y-xy²)/2 , or = x²y(1-y)/2.
  In the second form, we can see it's [x²][y(1-y)/2.], and as the things held constant pull out through the integrals, you can see it setting up to make E(X)E(Y) (or maybe E(Y)E(X)).
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HW:Expected values:
Ash p. 222 #4.
Probability handout:
   For the density of problem #17, find E(XY), and E(Y/X).  Also find E(X) and E(Y) separately from the marginals and check that E(XY) = E(X)E(Y)
   #18 c (You found the marginals last time.  Use them to find the E's)
   #19 c (do it by noting the form of the densities of X, and Y, and appealing to your list of means for known distributions)
A.  Here's one that's not independent:  f(x,y) = x+y, 0<x<1, 0<y<1.  It's a plane slicing through the origin. (See it in DPGraph--see Day 36 for where, how)
       a) Find  f_X(x).  Argue by symmetry that  f_Y(y) has the same form, and write it down.
       b) Multiply  f_X(x) f_Y(y) .  Are X and Y independent?  why not?
       c) Find E(X).  Find E(Y).   Find E(XY).
       d) cov(X,Y) = E(XY) - E(X)E(Y).  Find cov(X,Y).  Is it positive or negative?