## Week 2

This week we mainly went over the necessary concepts of probability as explained in the chapters 1 through 4 of the book. We began by defining the notion of a probability space $$S,P$$, and mentioned there are two families: discrete (where we could assume $$S\subset \mathbb{N}$$ and continuous (where we would assume $$S\subset \mathbb{R}$$). To describe the probability $$P$$ easily, we recalled the following theorem:

(Radon-Nykodim)> If $$S$$ is discrete, the function $$p: S\longrightarrow \mathbb{R}: x\mapsto P(x)$$ satisfies $$P(A)=\sum_{x\in A} p(x)$$.
If $$S$$ is continuous, there exists an essentially unique function $$f:S\longrightarrow \mathbb{R}$$ such that $$P(A)=\int_{A} f dx$$
the function above was called the density of the space.
. We also introduced the cdf $$F$$ of the space as $$F(a)=\sum_{x \le a } p(x)$$ or $$F(a)=\int_{x\le a}f(x)$$ in the discrete resp. continuous case.
We showed that given a probability space, one could make a new one in two major ways:
1. the product $$(S_1,P_1)\times (S_2,P_2)$$ with probability $$P(A\times B)=P_1(A)\cdot P_2(B)$$.
2. Given a function $$X:(S,P)\longrightarrow T$$ (called a random variable in the context of probability), one could "push $$P$$ forward" to a probability $$P_X$$ on $$T$$ defined as $$P_X(B)=P(X^{-1}(B))$$. This $$P_X$$ is the "distribution" of $$X$$
There are a numerous number of properties relevant to a random variable: we defined the independene of two sets, then of two families of sets (if each choice of two sets is in turn independent) and finally of two random variables $$X,Y:S\longrightarrow T$$ (when the families $$X^{-1}(A)_{A \in T}$$ and $$Y^{-1}(B)_{B\in T}$$ are in turn independent.
We also recalled that there are two interesting values associated to a probability space (and hence a random variable:
1. the mean $$\mathbb{E}[X]$$ defined as $$\mathbb{E}[X]=\sum xp(x)$$ or $$\mathbb{E}[X]=\int_S xf(x) dx$$ in the descrete (resp. continuous) case.
2. the variance $$\textrm{Var}[X]$$ defined as $$\mathbb{E}[(X-\mathbb{E}[X])^2]$$