Math 300 , Spring 2004, Day 33, F, April 23 Hit reload to get most current version

Review discrete situation, into continuous
  Density method:   f_Y(y) dy = Area = f_X(x) dx.  Solve for  f_Y(y)
  CDF method:  F_Y(Y < y_o) = P(Y < y_o) = P(g(X) < y_o).  Restate so it is a statement about P (?  X  ?).  Use the distribution of X, and whatever means is easiest to get this into a formula, either for  F_Y or for f_Y.
Homework questions?
Try again...

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Next:  Two random variables "jointly distributed".  Ash Ch. 5.  First 5.1 and 5.2
In practice: Two random variables X, Y)  measured on the same experiment.
We've looked at discrete already (balls and chips in urns, tables in two rooms in a restaurant, two dice).
Sample space:  points in x-y space.
Discrete (p. 180):  joint probability function p(x, y) = P(X = x and Y= y): lump of probability sitting on that spot.
    Probability of a set:  Sum the probabilities of the points in the set.
Continuous (p. 171 ff)  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.
Need calculus for that:
 The dA is a little region in x-y space--you chop the x-y plane into little dA's and sum the volumes over all the little dA's that are in R.
How to calculate?  Chop into rectangular dA's by chopping the x axis into dx's and the y axis into dy's and making the grid. Then you can sum them by first summing in (say) the x direction, and then summing those values in the y direction.
Discrete analog:
To find the probability of the whole space:
First sum across the x direction, get P(Y=1) = 7/20,  P(Y=2) = 6/20, P(Y=3) = 7/20,for each y.. Then sum over the y values, to get (7+6+7)/20 =1.

To find P(X + Y < 4):  In the yellow region, Sum across the x direction, get 4/20, 3/20, 3/20 for y = 1,2,3.  Then sum over the y values to get 10/20.
 

Integrals:

In the inner integral, the "other" variable acts like a constant.  Try it with f(x,y) = y

New HW Practice here: http://www.math.temple.edu/~cow/
Calculus Book III > 5.Integration > 2.Double Integrals >
Modules 1 and 2 Double integrals over rectangles  (Module 1 tells you the limits of integration and steps through.  Module 2 has you set the limits and do the whole thing.  Don't worry if you can't compute all the indefinite integrals, do enough to understand the pattern, and skip sin and cos if you want.) Do at least Module 1:  1, 10, 12,  (recall e^a+b = e^ae^b), Module 2: 1, 9
Module 3 Limits of double integrals--Figuring out the limits of integration.  Caution--some problems can be done easily both ways, and the package will allow you to do either.  But if there is only one way to do it without breaking it into 2 double integrals, it won't tell you you've chosen an impossible way--it'll just keep rejecting your answers. (e.g. Problem 1--if you (foolishly) try to integrate over x first, you need to break the area into 2 pieces, at the line y = 1.  The integration limits are "between y/2 and  y for 0<y<1,  and between y/2 and 1 for 1<y<2 ", which the program has no way of reading or understanding.)  Copy the sketch at the first step, to help keep track.
Don't worry about ones where you can't guess what the indefinite integrals are.
Do as many as you need to. At least 1, 5, 12. Ash pp. 165-69 is a good followup here.
Module 4 Changing the order of integration.  This requires figuring out from the integrals what the shape of the region is and the functions defining the borders.  Then switching, using the methods of module 3.  A good test for your understanding.  Try a few, at least 1, 4, 12.

This page belongs to Sally Sievers who is solely responsible for its content. Please see our statement of responsibility.