Recent content by zzzhhh

I have no clue what you are talking about either. More often than not, to understand a question is harder than to answer it in a haste. You should also learn it.

Definition of what? $s$ is the length of the arc, x in bold is the position vector. No definition of them can account for the second = in your reply. You must have mixed them up, so I said you are wrong.

Wrong.

This is excerpted from George E. Hay's "Vector and Tensor Analysis". The author gives the statement that the limit is equal to 1 without any explanation, perhaps because he thinks it does not belong to the contents of vector analysis. I can see it intuitively, but I want a rigorous mathematical...

I'm now reading "Vector and tensor analysis", Louis Brand, 1948, together with some other books that refresh my mathematical foundations.

please check you proof again. how did E[XY]E[XZ] = E[X^2YZ] imply E[X^2] = E[X]?

You are just listing some formulas in a textbook. It is neither a rigorous proof nor a counterexample. I think mathematical question is the most clear thing in the world -- it is just either true, if you can prove it, or false, if you can give a counterexample. Any effort to turn mathematical...

To be somewhat more specific, suppose X, Y and Z are all mutually independent random variables from uniform distribution on (0,1). I can find out that A=XY has a pdf f(a)=-ln a if 0<a<1 and 0 elsewhere, and so does B=XZ. Can you tell me what's the joint pdf of A and B f_AB(a,b), and show next...

I can here come up with an example of independence. we know sample mean and sample variance are independent for normal population. These two guys have many things in common -- all random samples, actually the definition of sample variance even contains sample mean, but they are indepemdent.

"This is why they are dependent." I still don't know why if a and b have common factor x then they are dependent (in terms of rigorous definition appeared in all probability theory textbook). OK, mathematics is not literature or philosophy. If you think A and B may be dependent, please come up...

No, b=az/y is not like the situation where in y = x + 2, y depends on x because 2 is a constant while z/y is a random variable. Considering the situation y=x+w, where all variables are independently random, does y depend on x? I think no, because y can take whatever value no matter what x is...

can't get it. what does "in terms of the simple explanation" mean? what does "that is ok" mean to get the joint pdf of A and B?

Supposing X, Y and Z are three mutually independent continuous random variable, is multiplication X*Y and X*Z still independent? If yes, please prove it; if no, please come up with a counterexample. Thanks a lot!

The exposition here is wrong, check the errata 1 of this book from the author's website ... oops!

This question comes from proof of Theorem 2.47 in Folland's "real analysis: modern techniques and their applications", second edition. In particular, the question lies in the inequalities in line 7 and 8 in page 76. The first equality is an application of measure property "continuity from...