# A Few Productive Days

I have had a few productive days, both in my studies, and in my life in general.

I am up to page 250 in Guo.  I read through the sections on the Beta, Weibull, Pareto, Lognormal, and Chi-square distributions, and did some problems, but did not put any particular effort into them.  When I am finished with the rest of the book, I will return and make a more thorough study.

What I did put effort into is the section on joint density and double integrations.  I have been doing these kinds of problems for a long time, and have been thrown off by more than a few of them, but have never really put any thought into a standard technique for solving them.

I could have just cracked a calculus book to find the same information.  First, I check my copy of Apostal (1961), but I only have volume I.  So then, I check Stewart (1999) and Larson (1990), which do have equivalent information.  Even easier, I could have spent a little more time with The Actex P/1 Study Manual (Boverman, 2010) which does have very specific ideas for settings up double integrations.

The important thing is to draw good diagrams.  It is easy to become complacent with the idea of solving mathematics problems completely symbolically.  A graph may not prove anything, but it can sure get you on the right track.

So, I am up to 466 memorization items.  One thing that I am memorizing is some nameless distributions that appear in old tests. One example is the distribution:

f(x) = 2x / L2, 0 < x < L

I have noticed that this distribution appears on many old problems.  If you check, it integrates to 1, so it is a pdf.  Well, it is easy to find formulas for E(X) and Var(X), and memorize them you would for any other named distribution.  Then, one does not need to perform time-consuming integrations at test time.

# Gamma Distributions

 Gamma Function: $\displaystyle \Gamma (\alpha)= \int_0^\infty e^{-y}y^{\alpha-1}dy$ Gamma Distribution: $\displaystyle \frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)}$ Gamma Distribution: $\displaystyle \frac{\beta^n x^{n-1}e^{-\beta x}}{(n-1)!}$ $\displaystyle \Gamma (n) = (n-1)!$ $\displaystyle \frac{x^{\alpha-1}e^{-\beta x}}{\theta^{\alpha}\Gamma(\alpha)}$ $\displaystyle \frac{x^{n-1}e^{-\beta x}}{\theta^n (n-1)!}$ $\displaystyle \Gamma (n+\frac{1}{2})\\ = 1\times3\times5\times, ..., (2n-1)\frac{\sqrt(\pi)}{2^n}$ $\displaystyle \beta e^{-\beta x} \frac{(\beta x)^{\alpha-1}}{\Gamma(\alpha)}$ $\displaystyle \beta e^{-\beta x} \frac{(\beta x)^{n-1}}{(n-1)!}$