HW Multivariate Continuous:
(repeated from Day 33, Ash 2b added.. Finish the remaining parts now)
A) Write up completely the work we did in class with g(x,y) = y, 0<x<4, 0<y<3;
Finding C =18 to
create density f(x,y) = g(x,y)/18, finding P(X+Y < 4), finding marginals.
DO on Handout "MultCont", #17 and #18 (p. 2): First,
Sketch the region of positive density of f, and do parts a.
Next: Also 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 and b graph the region of f,
and the region you want the prob. of. Next, 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).
Ash p. 181, #7. 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.)
Postpone
HW Multivariate Continuous, a few more on Independence:
A. On the Handout "MultCont", are the variables in problem 17 independent?
How about in 18?
Ash 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
= = = = = = = = = = = = = = = = = = = = = = = = = = = =
Two random variables X, Y "jointly distributed". Ash Ch. 5.
First 5.1 and 5.2
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.
Day 33
Homework questions: Last chance on
transformation of one random variable.
Jointly distributed: Discrete distributions, Continuous, drawing the regions.
Doing double integrals.
Day 33 for notes on continuous
Useful term from handouts: "support" of the distribution = region
in x-y plane that has non-zero probability or density = "universe" in
Ash.
Spent day on Day 33's
notes, and DP graph. The Uniform case below should be ok without lecturing (Ash
p.181 #7 above). Start
here WED
Special easy case:
If the distribution is Uniform over a region in the x-y plane, the f
= is a constant =1/(Area of the support region)
P (event B)= (Area of B)/ (Area of support region)--what
we did back in the beginning (Ash. pp. 31-2)
More on X, Y independent: p(x, y) = pX(x)pY(y),
f(x,y) = fX(x) fY(x)
for every pair (x,y) in the plane.
NOTE: X and Y canNOT be independent unless their joint support (region
where p or f is Not 0) is rectangular!
Because if support isn't a rectangle, there will be a pair (x,y) where X and
Y have positive probabilities/densities, but the pair has 0 prob/density.
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: "MultCont"
p. 2 # 19 factors to two
exponentials.
Next: Back to Ch. 7, Expected
values in continuous distributions
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