Math 300 Day39

Math 300 , Spring 2002, Day39, F, May 3Hit reload to get most current versionAfter class

Continuing with Two random variables X, Y "jointly distributed". Ash 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.
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.
"Marginal" probability/density function:

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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?
Continuous: same formulas as discrete, 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.
The conditional probability density of Y given 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, integrate (sum) over all the y's. The integral 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. So the conditional density is a "legal" density.

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 Integral of y f_Y|x(y|x_o) dy.
    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 (the abstraction from data), E(Y|x_o) can play that role.
             Find it for "all" x_o's, and graph it on the x-y plane.
Handout: Multivariate distributions of the Continuous Type: Example 3.7-2 ff.
We went through the handout, did not do the remaining examples in class.
--Find the conditional densities and the conditional expectations for the two functions above:
   See DPGraph pictures--slicing x will give shape of conditional density (files are labeled by formula)
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.

Astonishing(?) fact: IF E(Y|x) is linear in x, it will coincide with the (abstraction of) the familiar least squares regression line formula.
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New topic: Last semester we were told that the sum of two Normal random variables is normal.
Question: How would you find the distribution of W = X+Y?
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HW: Ash 8.1 covers conditional for continuous only. Does a good job.
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 had hoped to do this in class but didn't get there. The cardboard model.. Write it up.)
Ash, p. 249, #2, #5a, #3. (These don't need to be handed in--but DO understand them.)

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