1. ## distribution manipulation

I'm massively confused about what you can do to distributions formulas, and I haven't seen a nice collection of the rules anywhere.

So when you have something like $\displaystyle u \sim N(0, \sigma^2)$, I understand that you can say that $\displaystyle \frac{u}{\sigma^2} \sim N(0, 1)$ or that $\displaystyle u^2 \sim \chi^2$. But I don't know why these rules are true and how many of these rules exist. Can someone please point me to a useful resource? How can I manipulate these distribution expressions?

2. Originally Posted by garymarkhov
I'm massively confused about what you can do to distributions formulas, and I haven't seen a nice collection of the rules anywhere.

So when you have something like $\displaystyle u \sim N(0, \sigma^2)$, I understand that you can say that $\displaystyle \frac{u}{\sigma^2} \sim N(0, 1)$ or that $\displaystyle u^2 \sim \chi^2$. But I don't know why these rules are true and how many of these rules exist. Can someone please point me to a useful resource? How can I manipulate these distribution expressions?
This is a time consuming question to answer properly.

Most books on mathematical statistics will have something to say about it.

And a book I particularly like: The Algebra of Random Variables by M.D. Springer (most university libraries should have it).

I also suggest doing a Google search to find things like http://www.cs.unm.edu/~williams/cs530/gst2.pdf

And of course if you search the Advanced Probability and Statistics subforum at MHF you will find plenty of examples that use a variety of techniques.

3. Originally Posted by mr fantastic
This is a time consuming question to answer properly.

Most books on mathematical statistics will have something to say about it.

And a book I particularly like: The Algebra of Random Variables by M.D. Springer (most university libraries should have it).

I also suggest doing a Google search to find things like http://www.cs.unm.edu/~williams/cs530/gst2.pdf

And of course if you search the Advanced Probability and Statistics subforum at MHF you will find plenty of examples that use a variety of techniques.
Thanks for the book suggestion. I'll check it out.

Any chance you could suggest three rules that are most commonly used and that I might find useful?

4. Using moment generating/characteristic functions, which characterize a distribution & using the Jacobian transformation

5. Originally Posted by garymarkhov
Thanks for the book suggestion. I'll check it out.

Any chance you could suggest three rules that are most commonly used and that I might find useful?
1. Method of Distribution Functions.

2. Method of Transformations.

3. Method of Moment-Generating/Characteristic Functions.