random variable such that $\int_{\Omega}||X(\omega)||d\mathbb{P}(\omega)<\infty$ almost surely, the quantity $$\begin{aligned}
     \mathbb{E}(X) :=\int_{\Omega}X(\omega)d\mathbb{P}(\omega) =\int_{\mathbb{R}^n}d\mu_X(x)
     \end{aligned}$$ is called the expected value of $X$, where $||.||$ denote the euclidean norm on $\mathbb{R}^n$.

2.  In general, if $f : \mathbb{R}^n\longrightarrow \mathbb{R}^m$ is measurable and $\int_{\Omega}||f(X(\omega))||d\mathbb{P}(\omega)<\infty $ almost surely, then the qauntity $\mathbb{E}(f(X))$ define by $$\begin{aligned}
    \mathbb{E}(f(X)) :=\int_{\Omega}f(X(\omega))d\mathbb{P}(\omega)=\int_{\mathbb{R}^n}f(x)d\mu_X(x)\end{aligned}$$ is called expected value of $f(X)$.

**\[Independent random variables\]**

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space.

1.  Two elements $A$ and $B$ of $\mathcal{F}$ are independent if $$\begin{aligned}
    \mathbb{P}(A\cap B)=\mathbb{P}(A)\cap\mathbb{P}(B).\end{aligned}$$

2.  Two random variables $X_1$ and $X_2$ of $(\Omega, \mathcal{F}, \mathbb{P})$ are independent if for every choice of different borel sets $B_1$ and $B_2$ the following holds : $$\begin{aligned}
    \mathbb{P}(X_1\in B_1, X_2\in B_2)=\mathbb{P}(X_1\in B_1)\times\mathbb{P}(X_2\in B_2).\end{aligned}$$

The following proposition is from [@Oksa].

Two random variables $X_1$ and $X_2$ are independent if and only if for any measurable positive functions $f_1$ and $f_2$, the following equality holds $$\begin{aligned}
\mathbb{E}(f_1(X_1)f_2(X_2))=\mathbb{E}(f_1(X_1)) \mathbb{E}(f_2(X_2)).\end{aligned}$$

**\[Conditional probability\]**

For any event $A$ such that $P(A)>0$, the conditional probability on $A$ is the probability measure define by : $$\begin{aligned}
 \mathbb{P}(B/A) :=\dfrac{\mathbb{P}(A\cap B)}{\mathbb{P}(A)},\hspace{0.3cm} \forall B\in \mathcal{F}.\end{aligned}$$

### Conditional expectation

Let $X$ be a random variable such that $\int_{\Omega}|X(\omega)|d\mathbb{P}(\omega)<\infty$ almost surely. Let $\mathcal{G}$ a sub $\sigma$-algebra of 