Joint distribution of sequential variables

[Gear300]

Homework Statement

Let us choose at random a point from the interval (0,1) and let the random variable X1 be equal to the number which corresponds to that point. Then choose a point at random from the interval (0,x1), where x1 is the experimental value of X1; and let the random variable X2 be equal to the number which corresponds to this point.
(a) The marginal pdf f(x1) is intuitively uniform(0,1), and the conditional pdf f(x2|x1) is intuitively uniform(0,x1).
(b) What is the probability of X1 + X2 > 1?
(c) What is conditional mean E[X1|x2]?

Relevant Equations

Bayes' law: P(A|B) = P(A $\cap$ B)/P(B).

I am not too sure about parts (b) and (c). I guess it's reasonable to speak of a joint distribution of (X1,X2) after drawing both points.

 

[FactChecker]

You should show us some work on a homework problem before we can give you some hints.
You can start with drawing the possible area on a square with X1, X2 axes and see if that helps you to write down the equations needed.

 

[Gear300]

The set of possible points should be that half triangular area in the unit square where x2 < x1. But I don't think it is uniformly distributed, since otherwise the marginal distribution of X1 would not be uniform. (Haha, silly problem I know, but I had to ask since I'm not sure.)

 

[FactChecker]

Good point! For homework-type problems, you need to show your work before we are allowed to give guidance and make comments.

 

[pbuk]

You should show us some work on a homework problem before we can give you some hints.

You could try drawing the area that satisfies X1 + X2 > 1 on a graph with x2, p(X2=x2) axes and see if that helps.

 

[Gear300]

Sorry for the belated reply. I guess I could try simulating it to see if I find something. Do I even need to find the joint distribution to answer parts (b) and (c)?

 

[pbuk]

I guess I could try simulating it to see if I find something.

Simulation is a good way to verify an answer (I did just that here), but will not help you learn anything.

Do I even need to find the joint distribution to answer parts (b) and (c)?

No. For part (b) sketch the graph I suggested in #2. For part (c) think about another way of writing $E(AB)$.