Recent content by shoeburg

The first one, I believe. My book usually has it written as X dP, but what you have written in the first one is equivalent, correct? And then you just put the limit taking as n -> infinity before the integral. How do you know P({X>n}) --> 0? Is this true for all random variables with a finite...

You are right. Supposing I said, assume E(X) is finite. Does that take care of this counterexample and issue?

I do agree that the integral of anything over a set of measure zero is zero. However, if X takes the value 0 over a set B with P(B) > 0, than the expectation (integral) of X over that set B is zero. The concept that an integral over a set with probability zero is zero is certainly intuitive, it...

Let An be a sequence of sets indexed by n. As n goes to infinity, P(An) = 0. Let X be a random variable. Prove that the limit as n goes to infinity of the integral of X over An, with respect to probabiliity measure P, equals zero. I thought that the monotone convergence theorem applies to when a...

I'm having trouble working out a few details from my probability book. It says if P(An) goes to zero, then the integral of X over An goes to zero as well. My book says its because of the monotone convergence theorem, but this confuses me because I thought that has to do with Xn converging to X...

Let F be any distribution function. With either the indefinite integral, or taking limits at plus and minus infinity, is there an equivalent expression to ∫ F(x)dx ? Can we derive one? Thanks.

Yes, the Cantor distribution! Funny that you ask, I'm actually trying to work out a few problems based on the Cantor distribution right now. Wikipedia gives the Cantor distribution's expectation as 1/2 based on a symmetry argument. I was wondering if there is any other way to calculate it. I'm...

How about for a d.f. F that is singularly continuous (neither absolutely continuous nor discrete)?

Okay I see. Would this be the other definition: E(X) = ∫ X dF(X) = ∫ x f(x) dx, where dF(X) = (dF/dx)dx = f(x)dx. But this is assuming dF/dx exists, or perhaps I should say this is assuming dF/dx is useful. What if F is a singular continuous distribution?

How do I read, interpret the following definitions for the expectation of a random variable X? Assume the integral is over the entire relevant space for X. (1) E(X) = ∫ x dP (2) E(X) = ∫ x dF(X) If I asked you to calculate (1) or (2) for an arbitrary X, how does it look? My only other...

Whoops again I didn't see your second post. I'll try that. I really appreciate your help.

Whoops. Does it make sense to say that since B is the smallest sigma-algebra on A containing the singletons, then it is at most countable, because if it was uncountable it couldn't be the smallest?

Okay. Let B be generated by the singletons. Since B is a sigma-algebra, it is closed under countable unions, and since it must contain every singleton, there must then only be a countable number of singletons, implying A is countable.

Okay. I'm slowly picking up on this. Specifically, I'm asked to prove: Let B be the collection of all subsets of A (a sigma-algebra?). A is countable if and only if B is generated by the singletons. See my only experience with generators is from a couple abstract algebra courses I've taken, so...

Hello, I just started a prob theory class and I'm a total beginner. I can't find the definition of the following anywhere: 1) What does it mean for a set to be generated by the singletons? In other words, how do I show a certain set is generated by the singletons 2) Similarly, what does it...