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 HW: Found P( X+Y<4) by
integration.
Find the marginal distributions:
Discrete:
pX(x) = sum p(x,y)
down over all the y's, pY(y)
= sum p(x,y) across over all the x's.
Day
34 graph. Are X and Y independent? Check.
(Discrete handout: Sometimes careful algebra can give
formulas for marginals, from a formula p(x,y).)
Continuous:
In
class: fY(y) = 2y/9, 0<y<3
Should find 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: Download
DPGraph from source or from
my site . In Macmillan 110, Machine "Green" (I'll
try for its neighbors.) Easier? Copy InstallDPGraph, For 300
Class folder, Readme file to your disk.
Class Material Folder/ Math 300/ DPGraph Lab 04/.
Open DPGraph Read Me 04 file, read what to do. It
should run ok from in there; if not, install it and get it from the Program
menu. Files for HW in For 300 Class folder (Clicking
on a data file won't find the program unless it's in the Program menu,
but you can open any data file from within the running program.) Backup
folder contains copy of everything.
DPGraph images. Double click? Or start DPGraph
and open from within.
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. Show x(1+y)andx+y=1.dpg
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, 0<x<4,
0<y<3. oS3oS4
xyy/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]4 = (1/18) [128/3
] = 64/27
Note E(XY) = oS3oS4xyy/18
dxdy = oS32y2/9 [oS4
x/4 dx]dy =[oS4 x/4
dx][ oS32y2/9 dy] =
EX EY
--- --- --- --- --- --- --- --- ---
HW:
Handout on discrete: Finish 5.1-1, do 5.1-3.e,f
(read the solutions to a-d).Corrections, comments:
5.1-1c,f: There's not much of a double
sum to write when the probability wanted is only one point. What
I'm looking for is that you try to write double sums that parallel what
you do in writing double integrals, but in the sum case you can actually
check your work by brute force. Do what makes sense. 5.1-3.f :The
inner sum should be y=x+1 to y=x+4, not x+5. Substitute for
x = 1 and check that the inner sum sums over the right y's. Likewise for
x = 2. x=3. x=4.
A) Find fX(x) for the example above, where f(x,y) = y/18
on the rectangle 0<x<4, 0<y<3.
DO Handout on Continuous "Probability", #17 and #18 (p. 2):
For the last assignment, you Sketched the region of positive density of
f, and did parts a.
Now: determine their marginal densities
and whether X and Y are indep.for 17,18,19, and finish
17, do 18b, 19b.
17, 18, 19 are in DPGraph
files to be viewed.(the z axis is distorted
to get a better picture.)
Ash p. 181, #2a For last assignment, graphed the region of f,
and the region you want the prob. of. Now finish it, do 2b..
(Hint: p. 164 has pictures for min and max--but in my book
the text refers to the wrong pictures (the captions are right),
#7. (Did already?)Do it
by graphing the regions, and since all pairs x,y are equally likely, find
the prob's by proportional areas (cf. pp. 31-2, and #7 p. 35.)
Independence:
p. 191, #5. For each part: If they are independent, what are
the marginals?. Sketch the regions where f is positive, before bothering
to look for the marginals.
#9
Read New Handout "Expected values",
and example 7 (not about expected values) and check the answer as the handwritten
part says.
You will want the ones from the Continuous
"Probability" handout to use as reference in the following problems.
Expected values: Ash p. 222 #4. (One-variable, good review.)
Postpone:
Probability handout:
For the density of problem #17, find E(XY), and E(Y/X).
#18 c (You found the marginals above. 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)
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