This book provides technical support for students in finance. It reviews the main probabilistic tools used in financial models in a pedagogical way, starting from simple concepts like random variables and tribes and going to more sophisticated ones like conditional expectations and limit theorems. Many illustrations are given, taken from the financial literature. The book is also a prerequisite for “Stochastic Processes for Finance” published in the same collection.

1. Probability spaces and random variables
   1. Measurable spaces and probability measures
   2. Conditional probability and Bayes theorem
   3. Random variables and probability distributions
2. Moments of a random variable
   1. Mathematical expectation
   2. Variance and higher moments
   3. The vector space of random variables
   4. Equivalent probabilities and Radon-Nikodym derivatives
   5. Random vectors
3. Usual probability distributions in financial models
   1. Discrete distributions
   2. Continuous distributions
   3. Some other useful distributions
4. Conditional expectations and Limit theorems
   1. Conditional expectations
   2. Geometric interpretation in L2 (O, A, P)
   3. Properties of conditional expectations
   4. The law of large numbers and the central limit theorem
5. Bibliography