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.
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.
Integrals:
In the inner integral, the "other" variable acts
like a constant. Try it with g(x,y) = y, 0<x<4,
0<y<3
Find area under the curve: Dividing g by this will give a function
with area 1. Let f(x,y) = Cg(x,y) = Cy as above.C
= 1/18
Monday: Find P( X+Y<4) by integration.
Find the marginal distributions:
fY(y)
= 2y/9, 0<y<3
Should have found for hw that fX(x)=
1/4, 0<x<4.
Are X and Y independent? Check.
NOTE: X and Y canNOT be independent unless their joint support
(region where p or f is Not 0) ("universe") is rectangular!
Separable joint densities (p. 185-6) X and Y are independent
If and Only If:
The universe is rectangular, and f(x,y) can be written as (a
function only of x) times (a function only of y).
Factoring the constant part of f(x,y) so the right amounts attach to
the x-function --to give a legal fX(x),
and to the y function--to give a legal fY(y)--can always be
done. Example: Prob. handout p. 2 # 19 factors to two exponentials.
3-dimensional graphs of HW problems: In Macmillan 110,
Machine "Black" (and "Gray" if I can get to it.)
Class Material Folder, Math 300. Open Read Me file, read what
to do.
DPGraph images. Double click? Or open DPGraph Viewer.
Turn image around with arrow keys, shrink/stretch with
PageDown/ PageUp. Scrollbar menu item, slice x, y, or z, then use
right scrollbar to move slice.
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.
--- --- --- --- --- --- --- --- ---
HW:
Handout on Probability(p. 2),#17
Finish, #18 do b, #19 do b : Finish or do. All are in DPGraph files
to be viewed.(the z axis is distorted
to get a better picture.)
Ash p. 181, #2a finish it. (Hint--p. 164 has pictures
for min and max--but in my book the text refers to the wrong pictures (the
captions are right)
p. 191, #5. If they are independent, find the marginals.
Sketch the regions where f is positive, before bothering to look for the
marginals. #9
Since people are not handing
in much anyhow, if you do the above, hang on to the ones from the probability
handout to use as reference in the following problems.
Rest postponed:
Expected values: Ash p. 222 #4.
Probability handout:
For the density of problem #17,
find E(XY), and E(Y/X).
#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)
| Sievers home |
Math300-Sp02/Day35.htm
|
3pm
|
4/24/02 |