### From Actuarial Outpost Wiki

**Probability**

*see also: discussion forum for exam-1/p:[1]*

The examination for this material consists of 3 hours of multiple-choice questions and is identical to CAS Exam 1.

The purpose of this course of reading is to develop knowledge of the fundamental probability tools for quantitatively assessing risk. The application of these tools to problems encountered in actuarial science is emphasized. A thorough command of probability topics and the supporting calculus is assumed. Additionally, a very basic knowledge of insurance and risk management is assumed. A table of values for the normal distribution will be included with the examination booklet.

## LEARNING OUTCOMES

Candidates should be able to use and apply the following concepts in a risk management context:

- General Probability

- Univariate probability distributions (including binomial, negative binomial, geometric, hypergeometric, Poisson, uniform, exponential, chi-square, beta, Pareto, lognormal, gamma, Weibull, and normal).

- Multivariate probability distributions (including the bivariate normal)
- Joint probability functions and joint probability density functions
- Joint cumulative distribution functions
- Central Limit Theorem
- Conditional and marginal probability distributions
- Moments for joint, conditional, and marginal probability distributions
- Joint moment generating functions
- Variance and measures of dispersion for conditional and marginal probability distributions
- Covariance and correlation coefficients
- Transformations and order statistics
- Probabilities and moments for linear combinations of independent random variables

### Suggested Texts

There is no single required text for this exam. The texts listed below may be considered as representative of the many texts available to cover material on which the candidate may be examined.

Not all the topics may be covered adequately by just one text. You may wish to use more than one of the following or other texts of your choosing in your preparation. Earlier or later editions may also be adequate for review.

- A First Course in Probability (Sixth Edition), 2001, by Ross, S.M., Chapters 1–8.

- Fundamentals of Probability (Third Edition), 2005, by Ghahramani, S., Chapters 1–11.

- John E. Freund’s Mathematical Statistics with Applications (Seventh Edition), 2004, by Miller, I., Miller, M., Chapters 1-8.

- Mathematical Statistics with Applications (Sixth Edition), 2002, by Wackerly, D., Mendenhall III, W. Scheaffer, R., Chapters 1-7.

- Probability for Risk Management, 1999, by Hassett, M. and Stewart, D., Chapters 1–11.

- Probability: The Science of Uncertainty with Applications to Investments, Insurance and Engineering 2001, by Bean, M.A., Chapters 1–9.

## Study Notes

### Discrete Distributions

- PLEASE NOTE:
- In the formula for Variance of Uniform you have a mistake, the term you are subtracting should be <math>\left( \frac{\sum_{i=1}^{n} x_i}{n} \right) ^2 </math> I believe. Also, at least for me, where it says <math> n \in ...</math> it's coming out weird, is that supposed to be <math> n \in Z^+</math> ?
- In general (continuous case) the variance of U[a,b] is <math>\frac{\left( b - a \right) ^2}{12}</math>. This follows immediately from the fact that the variance of U[0,1]=1/12 and a U[a,b] is a shifted and rescaled U[0,1]. The discrete case has a correction factor and so it isn't just n^2/12. the variance for discrete is is <math>\frac{n^2 - 1}{12}</math>

### Discussion Forum Exam P/1 Notes Depository

Download more notes in .doc, .pdf and other formats here

### Other Notes Info

SNs for the Preliminary Education examinations and Course 6 are available on the SOA Web site under Education and Jobs/Candidate and Exam Information/Spring Exam Session/Spring 2005 Basic Education Catalog – Study Notes Information. Hard copies may be purchased by using the Study Note and Published Reference order form in the back of the printed catalog or by downloading the form from the Spring Exam Session Web page.

Code Title

P-05-05# P Introductory Study Note

P-09-05# P Sample Exam Questions and Solutions

P-21-05# Risk and Insurance