Math 300 , Spring 2004, Day36, F, April 30 Hit reload ...Corrected (I hope) after class

Continuing with Two random variables X, Y "jointly distributed".  Ash Ch. 5.  First 5.1& 5.2 Next 7.1 again, then 8.1
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.
"Marginal" probability/density function:
      Function for just X (not "looking at" y):   p_X(x) or f_X(x)
                    or just Y (not "looking at" y):   p_Y(x) or f_Y(y) .
X and Y INDEPENDENT:  p(x, y) =  p_X(x)p_Y(y),  f(x,y) = f_X(x) f_Y(y) for every pair (x,y)
 

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.
 

- - - - - -  - - - - - -
  HW questions?   Probability Handout problem #17:  Density f(x,y) = Cx(1+ y), 0<x<1, 0<y<2.
    C = 1/2.  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.

Find E(XY) for f(x,y) = y/18,  0<x<4, 0<y<3. E(XY)=  _oS³_oS⁴xyy/18 dxdy = (1/18) _oS ³ [(1/2) x²y² _x=o]⁴dy
          = (1/18) _oS ³ [8y²  - 0]dy = (1/18)  [8y²/3 _y=o]³ = (1/18)  [127 8· 27/3] = 4
Note E(XY) = _oS³_oS⁴xyy/18 dxdy = _oS³2y²/9 [_oS⁴x/4 dx]dy =[_oS⁴ x/4 dx][ _oS³2y²/9dy] = EX EY

Now begin E(XY) for the density f(x,y) = Cx(1+y), 0<x<1, 0<y<2.  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)).
Got to here:
Conditional distributions: Assuming a particular x value is true/known (x_o),  what is the distribution of Y?
   If you have 4 chips, red on one side (X) and blue on the other (Y),
x red/y blue   1/4  2/2  2/4  3/4   and you choose a chip & see that the red side is 2, what is now the probability distribution of the numbers on the blue side?
Put in usual 2-dimensional x-y grid.
Assuming x_o known, what is the distribution of Y?. Look just at the column for x_o.
Can do the other way: assuming y_o known, what is the distribution of X?.Look just at the row for y_o.
  Discrete: P(Y= y | X = x_o) = P(Y = y & X = x_o)_/P(X = x_o)   (Old conditional probability, in new clothes)
     Changing to distribution language:  The conditional probability (density) of Y given x_o:
           P(Y= y | X = x_o) =  f_Y|x(y|x_o) = f(x_o, y)/f_X(x_o)
   For a single particular number x, you can plug in that number, and then everything is in y.
        For a particular number x, sum over all the y's.  The sum over the y's of f(x_o,y) = f_X(x_o).
             So the sum over all the y's of f_Y|x(y|x_o) = f_X(x_o)/f_X(x_o) = 1.
    Look at handout: Conditional distributions: Discrete.
 --- --- --- --- --- --- --- --- ---
HW:Expected values:
Ash p. 222 #4.(assigned last time)
Probability handout:
   For the density of problem #17, find E(XY)(started in webpage above), 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 35 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?
Postpone conditional.  Read first side of Conditional (discrete) handout.
Conditional:
B. For the chips above, Find f_Y|x(y|x_o) when x_o = 3. Find f_X|y(x|y_o) when y_o = 4