Okay, since I am just getting started here, things will be a little chaotic as I randomly fill in background information.
My studies today:
- Did Mnemosyne repetitions for 40 minutes in a.m.
- Worked from page 123 to page 143 in Guo.
- Added 16 items to Mnemosyne. Up to 416 now.
- Worked a few problems from SOA sample problems.
Here is the logic of my current study strategy. I have spent much time with all levels of statistics books, and with study manuals specifically aimed at test p. I have also spent many hours with sample SOA problems. Hence, I have a fairly good grasp of both theory and practice. At this point, I am interested in filling in my weak areas of theory, expanding the base of problem types which I am very comfortable with, and firmly memorizing formulas and relations that are within the test p domain.
With those goals in mind, my study choices make more sense. The book by Guo is Deeper Understanding, Faster Calculation–Exam P Insights & Shortcuts. It is available by download for $85 from all the usual places. I have to think twice about an 85 dollar book, but I am now very interested in streamlining my solutions to common problem types. I have devised my own streamlined solution to some problems, but I could see that Guo has many good ones in his book, based upon the free chapter available online. For instance, Guo advises memorizing the integrals for any integration by parts that might very likely appear on the test. Integration by parts take precious time, and are pitfalls for arithmetic errors (all of those negatives!). So, I am working my way front to back through Guo, and mining it for good stuff. Anything good gets transformed into several simple question and answers, then loaded into Mnomosyne, a spaced repetition memorization program.
Guo is not a book I could have started with, but was just the thing that I needed at this point in my studies. I have got by in mathematics with whatever I remember from working through books, working through problems, and writing about math. To become absolutely fluent in the manner demanded by the actuarial tests, however, one must spend time committing important things to memory. Take for example, the order statistics formulas:
fx(k) = n! / (k-1)! (n-k)! F(x)k-1 (1-F(x))n-kf(x)
I could work order stat problems all day and not remember that formula. Yet without it, even the very simple order problems likely to be on the test would be unsolvable in an acceptable amount of time.
While I am thinking about streamlined solutions, I need to share a simple one. We often see problems such as:
“The failure time of a machine has an exponential distribution with a median of 5. What is the probability that the machine runs for more than 6 hours?”
The solution to this problem is simple, yet has unneeded steps:
- First solve for λ. F(x)=0.5, so 1-e-5λ=0.5.
- Take log. 5λ = ln0.5
- λ = 0.6931/5 = 0.1386
- Now use this value of λ to find e-0.1386*6 = 0.4353
After working many problems like this one for the last year, it suddenly dawned on me that I needed an efficient way to get from the median to the mean of an exponential distribution. It took me all of five minutes to find that the median = the μ * ln2, and μ = the median / ln2. Of course λ is simply 1/μ, but μ is much more intuitive. This was simple mathematical relation that was bound to prove to be useful.
My next step was to turn this fact into a set of simple questions and answers which I could then add to my list of things to memorize.
- Exponential distribution, median = μ ∙ ???
- Exponential distribution. μ = median / ???
- Exponential distribution. Mean is ??? than the median.
- greater. μ = median / ln2 incorrect!!!
- Exponential distribution. Median is ??? than the mean.
- less. Median = μ * ln2 incorrect!!!
- Exponential distribution with μ = 1. Median = ???
- ln2 = 0.693157
- Exponential distribution with μ = 20. Median = ???
- 20 ∙ ln2 = 13.86
- Exponential distribution with μ = 250. Median = ???
- 250 ∙ ln2 = 173.29
- Exponential distribution with μ = 400. Median = ???
- 400 ∙ ln2 = 277.26
- Exponential distribution with median = 1. μ = ???
- 1 / ln2 = 1.443
- Exponential distribution with median = 5. μ = ???
- 5 / ln2 = 7.2135
- Exponential distribution with median = 600. μ = ???
- 600 / ln2 = 865.6
After writing these questions, I loaded them all into Mnemosyne.
The above questions reflect my current method of formulating memorization items, which is to break an item down into several smaller items, some of which stress concept, some relations to other concepts, and some numerical examples.
More to come on all of these topics!