I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 A. Definition of normal variable.
 B. Linear transformation of random variables.
 C. Multivariate normal distribution. Choleski decomposition.
 D. Calculus of normal variables.
 E. Central limit theorem (CLT).
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Linear transformation of random variables.

e introduce the notation to describe the statement "random variable has distribution ".

Proposition

Let be a vector of random variables and for some function . Set for some deterministic square matrix and vector . If then

Proof

For any domain of the -space we can write We make the change of variables in the last integral.

 (Linear transformation of random variables)
Hence

The linear transformation is distributed as . The was defined in the section ( Definition of normal variable ).

For two independent standard normal variables (s.n.v.) and the combination is distributed as .

A product of normal variables is not a normal variable. See the section ( Chi squared distribution ).

 Notation. Index. Contents.