Math 300 , Spring 2002, Day40, M, May 6Hit
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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.
fX(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
fX(x)
= 2x, 0<x<1. fY(y) = (1+y)/4, 0<y<2.
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Conditional distributions: Assuming
a particular x value is true/known (xo), what is the distribution
of Y?
Continuous:
same formulas as discrete, integration instead of sums.
Think
of slicing through the joint density at xo--look at f(xo,y)--the
slice.
The
conditional density fY|x(y|xo) has the shape
of the slice, but the slice may not have area 1.
If we divide by fX(xo), it will.
The conditional probability density of Y given
xo:
fY|x(y|xo) = f(xo, y)/fX(xo)
Conditional Expectation E(Y|xo):The
mean y value, when x is a particular fixed value xo.
Treat xo
as a constant and find the Integral
of y fY|x(y|xo)
dy.
Remember
"Regression problem" in statistics: For a particular xo,
predict the "best" y-value.
("best" in some sense or other--average, typical...)
In probability
(the abstraction from data), E(Y|xo) can play that
role.
Find it for "all" xo's, and graph it on the x-y plane.
--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.
fX(x) = 2x, 0<x<1.
fY(y) = (1+y)/4, 0<y<2.
f(x,y) = x+y, 0<x<1, 0<y<1.
fX(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? Try it
on discrete: from handout.
f(x,y) = (x+y)/21, x = 1,2,3, y= 1,2. (recall sum of 2 dice)
Add on lines x+y=w, or x = w-y.
(Ch. 6.1 does more of this, including
continuous case. We won't.)
Another new topic: Joint normal distribution:
(handout)
Correlation coefficient rho = cov(X,Y)/stdev(X)stdev(Y)
(cf. sample correl. coeff.)
This is also a parameter
in the joint normal. Not coincidentally, it turns out to be the correlation
coefficient for X,Y.
Level curves are ellipses.
E(Y|x) is line
that bisects "vertical" distances inside ellipses.
DPGraph pictures.
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HW: Read Normal handout. Catch
up?
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