Independence

Def)
$X_1,X_2$ are independent $<=>$ $\left\{\begin{matrix} f(x_1,x_2) = f(x_1)f(x_2)\\ p(x_1,x_2) = p(x_1)p(x_2) \end{matrix}\right.$ for all $x_1,x_2$

Corollary)
If $X_1,X_2$ are indep.,
① $P(X_1 \in A, X_2 \in B) = \int_{B} \int_{A} f(x_1,x_2) dx_1 dx_2 = \int_{A} f(x_1) dx_1 \int_{B} f(x_2) dx_2 = P(X_1 \in A) P(X \in B)$

② $f(x_2|x_1) = \frac{f(x_1,x_2)}{f(x_1)}=\frac{f(x_1)f(x_2)}{f(x_1)}=f(x_2)$

Theorem) factorization theorem
Let $X_1,X_2$ have supports $A,B$ respectively,
$X_1,X_2$ are independent $<=>$ $\left\{\begin{matrix} f(x_1,x_2) = g(x_1)h(x_2)\\ p(x_1,x_2) = g(x_1)h(x_2) \end{matrix}\right.$

where $g(x_1)>0,x_1 \in A$ and $h(x_2)>0,x_2 \in B$
pf)

Theorem)
The followings are equivalent
① $X_1,X_2$ are independent
② $F(x_1,x_2)=F_{X_1}(x_1) F_{X_2}(x_2)$
③ $P(a < X_1 < b, c < X_2 <d) = P(a < X_1 < b) P(c < X_2 < d)$
④ $M_{X_1,X_2} (t_1,t_2) = M_{X_1}(t_1) M_{X_2}(t_2)$
pf)

Theorem)
If $X_1,X_2$ are independent,
$E[u(x_1)v(x_2)] = E(u(x_1))E(v(x_2))$

✔︎ 역은 성립하지 않음!