Math 300 , Spring 2002, Day38, W, May 1Hit reload to get most current versionAfter Class

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 Visualizations:  (Mac 110 Black or Gray: Class Materials/Math 300/For300 Class (DPGraph program is in Math 300 folder)
f(x,y) = x+y, 0<x<1, 0<y<1.    Cardboard model.
     f_X(x) = x + 1/2, 0<x<1
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(y) = (1+y)/4, 0<y<2.
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Conditional distributions: Assuming a particular x value is true/known (x_o),  what is the distribution of Y?
   If you have chips, red on one side (X) and black on the other (Y),  and you choose one & see that the red side is 2, what is now the probability distribution of the numbers on the black side?
Can do the other way: assuming y_o known, what is the distribution of X?.
  Discrete:  = 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.
  Conditional Expectation E(Y|x_o):The mean y value, when x is a particular fixed value x_o.
      Treat x_o as a constant and find the sum of all   y f_Y|x(y|x_o) terms.
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HW Day 38  Conditional Distributions (discrete) handout:
  problem 2.10-1
  A. (Show that X, Y independent -->f_Y|x(y|x) = f_Y(y); that is, knowing X gives no info about Y)
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Friday we'll continue with:
 Continuous:  same formulas, integration instead of sums.
      Think of slicing through the joint density at x_o--look at f(x_o,y)--the slice.
      The conditional density f_Y|x(y|x_o) has the shape of the slice, but the slice may not have area 1.
          If we divide by  f_X(x_o), it will.
handout: Multivariatedistributions of the Continuous Type: Example 3.7-2 ff.
Remember "Regression problem" in statistics:  For a particular x_o, predict the "best" y-value.
           ("best" in some sense or other--average, typical...)
In probability, E(Y|x_o)  can play that role.  Find it for "all" x_o's, and graph it on the x-y plane.
--Find the conditional densities and the conditional expectations for the two functions above:
f(x,y) = x(1-y)/2, 0<x<1, 0<y<2.  f_X(x) = 2x, 0<x<1.   f_Y(y) = (1+y)/4, 0<y<2.
f(x,y) = x+y, 0<x<1, 0<y<1.  f_X(x) = x + 1/2, 0<x<1
Another? f(x,y) = 2e^-x-y , y>x, x>0.
     Caution: if support of f(x,y) isn't rectangular, you have to keep track of possible values of x and y.
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HW:  Ash 8.1 covers conditional for continuous only.  Does a good job.
Conditional Distributions (discrete) handout:
  problem 2.10-1
  A. (Show that X, Y independent -->f_Y|x(y|x) = f_Y(y); that is, knowing X gives no info about Y)
Multivariate ...Continuous handout:
  problem 3.7-6
  B. (this is checking the equation in the middle of the second page of the handout.)
  C. (I may have started or even completed this in class.  Write it up.)
Ash, p. 249, #2, #5a, #3.  (These don't need to be handed in--but DO understand them.)