Lecture 1

Lecture

• January 6, 2025

Terminology

• Discrete Time: $T=\mathbb{N},\mathbb{Z},\text{ fininite set}$
• Continuous Time: $T=[0,\infty ),[0,1],\cdots$
• Discrete Space: $S:=\text{ a finite set },\mathbb{N}$
• Continuous Space: $S=\mathbb{R},{\mathbb{R}}^d,\cdots$

Review

Conditional Probability

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

Law of Total Probability. If $\{A_i\}$ is a partition of space $B$,

$$P(B)=\sum _{j}P(B|A_j)P(A_j)$$

Bayes Rule

$$P(A_i|B)=\frac{P(B|A_i)P(A_i)}{{\sum }_{j}P(B|A_j)P(A_j)}$$

Independence of 2 events

$$P(A\cap B)=P(A)P(B)⟺A,B\text{ are independent}$$

Distribution Function:

$$F_X(x)=P(X\le x)$$

• Discrete RV: $P_X(k)=P(X=k)$
• Continuous RV: $f_X(x)$ where $P(X\in A)={\int }_A{f}_X(x)dx$

Expectation

$$E[x]=\sum _i{x}_i{P}_X(x_i)$$ $$E[x]=\int x{f}_X(x)dx$$

Variance and Covariance

$$\text{Var}(X)=E[(X-E[X]{)}^2]=E[X^2]=(E[X]{)}^2$$ $$\begin{array}{rl}\text{Cov(X, Y)}& :=E[(X-E[X])(Y-E[Y])]\\ & :=\text{E}XY-\text{E}X\text{E}Y\end{array}$$

We say $X,Y$ are uncorrelated if $\text{Cov(X,Y)}=0$. In particular

$$X,Y\text{ independent}⟹\text{Cov}(X,Y)=0$$

other direction is not true in general.

Stochastic Process