# Probability and Stochastic Processes - Rome 2017

Last update: Oct 19, 2017.

What is needed to prepare for PART I:

• You need to understand your own notes taken in class !! This is the primary material for the course. All other references are useful supplementary material. For abbreviations, see below.
• [ER] is a reference where the topic of probability is treated at a very simple level You should try to do some practise on the exercises contained in this book
• [R] is more advanced. Many also find it clearer (less words, more rigour). Here you will find more information. Some of the arguments treated in class can be found here to be treated more widely (e.g. moments)
• [CA] This is a very friendly reference and a pleasure to read. Not strictly needed for the exam if you have your own notes and [ER]+[R], but helps intuition quite a lot !
• [GS] This reference has a slightly different order of exposition aimed at defining stochastic processes. Reading may be beneficial to students planning to attend also part 2.
###### Abbreviations:

[ER] Evans M.J. and Rosenthal J.S., Probability and Statistics The Science of Uncertainty, Freeman.

[R] Rohatgi K.L. Introduction to Probability and Mathematical Statistics. Wiley.

[CA] Chung K.L. and AitSahlia F., Elementary Probability Theory, Springer.

[GS] Grimmett, G. and Stirzaker D. Probability and Random Processes, OUP.

Notice. Point-by-point references to Rohatgi will be updated as soon as possible, but you may start looking at the corresponding Sections in the book by yourselves.

#### Topics treated so far.

1) Probability models

• Definition and properties of space of events, sigma algebras and probability measure (ER 1.1-1.3; CA 1-2.3; GS 1.1- 1.3)
• Conditional probability and independence of events (ER 1.5 - 1.7; CA 2.4-2.5; GS 1.4- 1.7)

2) Random variables and distributions

• Definition of random variables (ER 2.1; CA 4.1-4.2 and Appendix 1; GS 2.1, 2.3)
• Distribution of random variables (ER 2.2; CA 4.3 except expectations)
• Discrete and continous distributions (ER 2.3-2.5, CA 4.4- 4.6; GS 3.1, 3.5, 4.1, 4.4)

3) Functions of random variables and generating functions

• One-dimensional change of variable (ER 2.6; GS 4.7)
• Expectation (ER 3.1- 3.3, CA 6.1 - 6.2)
• Moments (R, to be updated)
• Generating functions (ER 3.4, until p. 159, CA 6.5; GS 5.1)