Q:

Bivariate and Multivariate Probability Distributions

For my homework for Bivariate and Multivariate Probability Distributions section, I encounter the terms joint density, joint distributed random variable, joint probability, uniform distribution, when a problem is of the format "Let Y1 and Y2 have the (ONE OF THE FOUR TERMS) function given by f(y1,y2) = SOME FUNCTION in the domain {0<= y1 <= y2 <=1}". I struggle to learn in a class environment for math classes, so I usually learn everything two weeks before my test either myself, or from my professor by doing couple of questions, so I am quite lost in this section. The questions typically ask me after providing this information to:
A. sketch the domain (I think I'll be able to do that). 
B. show significant steps of integration that verify the constant (I can't really provide    this one because I'd have to give a particular problem, so it's okay if you ignore it)
C. Find P(Y1<1/3, Y2 >= 1/2) (the numbers are different per problem)
D. Find P(Y1+Y2 >= 1) 
Can you guys explain to me how from the four different terms I would approach each problem? (Like what would I be doing different for the four)
Also, can you give me general instructions on how to the letters? Thank you very much. 

A:

Well, if the question is phrased as you state, then they can really only have a joint density, as the other terms dont really fit your example sentence. I can offer you some general instructions for each letter:
B) Its just asking you to verify that the double integral of the density over the domain is equal to 1 over the constant, whatever it is, so the entire density integrates to 1. The only challenge will be if it a tricky integral.
C) and D) both of these are defining a new domain of integration for your given density. Whereas in B) you integrated over the entire domain, in C and D, you will be integrating over only part of the domain. In particular, you need to integrate over the following domains:
C) Over the region defined by $Y_1<a,Y_2\geq b$ given the general restritions on the variables you have provided, i.e, $\{0 \leq y1 \leq y2 \leq1\}$
D) Same thing but your domain of integration is defined as the region above the line $Y_1+Y_2=1$ and within the general bounds on the variables.

