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[수리통계학 I] 8, 9강 요약 정리

[수리통계학 I] Conditional Distribution ~ Extension

부산대학교 김충락 교수님의 2020년 1학기 KOCW 강의를 들으며 요약하는 글입니다.

$p_{X_2 \vert X_1}(x_2 \vert x_1) = \frac{p_{X_1}(x_1)}{p_{X_1, X_2}(x_1, x_2)}$ : conditional pmf of $X_2$ given $X_1= x_1$
$f_{X_2 \vert X_1}(x_2 \vert x_1) = \frac{f_{X_1}(x_1)}{f_{X_1, X_2}(x_1, x_2)}$ : conditional pdf of $X_2$ given $X_1= x_1$
분자: joint / 분모: marginal

$P(a < X_2 < b \vert X_1=x_1) = \int_{a}^{b} f(x_2 \vert x_1) \, dx_2$
conditional mean of $u(X_2)$: $E[u(X_2)\vert x_1]= \int_{-\infty}^{\infty} u(x_2)f(x_2\vert x_1) \, dx_2$
conditional var. of $X_2$: $Var(X_2\vert X_1)= E[\lbrace X_2 - E(X_2\vert x_1)\rbrace^2 \vert x_1] = E({X_2}^2 \vert x_1) - E^2(X_2 \vert x_1)$

Double expectation theorem: $E[E(X_2 \vert X_1)] = E(X_2)$
$Var[E(X_2\vert X_1)] \leq Var(X_2) = Var[E(X_2\vert X_1)] + E[Var(X_2\vert X_1)]$

$Cov(X, Y) := E[(X_1-\mu_1)(Y-\mu_2)] = E(XY) - E(X)E(Y)$ : covariance(공분산) between X and Y
Corr. coef.: $\rho := \frac{Cov(X, Y)}{\sigma_1\sigma_2} = \frac{E[(X-\mu_1)(Y-\mu_2)]}{\sigma_1\sigma_2}$
if $E(X\vert Y)$ is linear in $X$
- $E(Y\vert X) = \mu_2 + \rho\frac{\sigma_2}{\sigma_1}(X-\mu_1)$
- $E[Var(Y\vert X)]= {\sigma_2}^2(1-\rho^2)$

$X$와 $Y$가 독립이다: $f_{X,Y}(x,y)= f_X(x)f_Y(y)$
$X$ and $Y$ is indep. $\Leftrightarrow f_{X,Y}(x,y)= g(x)h(y)$, $g(x)$와 $h(y)$가 각각 x와 y에 대한 함수
$X$ and $Y$ is indep. $\Leftrightarrow F_{X,Y}(x,y) = F_X(x)F_Y(y)$
$X$ and $Y$ is indep. $\Leftrightarrow P(a < X \leq b, c < Y \leq d) = P(a < X \leq b)P(c < Y \leq d)$
$X$ and $Y$ is indep. $\Rightarrow E[u(X)v(Y)] = E[u(X)]E[v(Y)]$
$X$ and $Y$ is indep. $\Leftrightarrow M(t_1, t_2)= M(t_1,0)M(0,t_2)$

$\mathbf{X} = (X_1, \cdots, X_n)^T$: n-dim random vector

$F_{\mathbf{X}}(\mathbf{x}) = P(\mathbf{X} \leq \mathbf{x}) = P(X_1 \leq x_1, \cdots, X_n \leq x_n)$ : joint cdf
$Y= u(X_1, \cdots, X_n) \Rightarrow E(Y)= \int \cdots \int u(x_1,\cdots,x_n)f_{X_1,\cdots,X_n}(x_1,\cdots,x_n) \, dx_1\cdots dx_n$
- $f_{2,\cdots,n\vert1}(x_2,\cdots,x_n \vert x_1) = \frac{f_{X_1,\cdots,X_n}(x_1,\cdots,x_n)}{f_1(x_1)}$
- $f_{1\vert 2,\cdots,n}(x_1 \vert x_2,\cdots,x_n) = \frac{f_{X_1,\cdots,X_n}(x_1,\cdots,x_n)}{f_{X_2,\cdots,X_n}(x_2,\cdots,x_n)}$

$E[\mathbf{A}_1\mathbf{W}_1 + \mathbf{A}_2\mathbf{W}_2] = \mathbf{A}_1E[\mathbf{W}_1] + \mathbf{A}_2E[\mathbf{W}_2]$
$E[\mathbf{A}_1\mathbf{W}_1 \mathbf{B}] = \mathbf{A}_1E[\mathbf{W}_1]\mathbf{B}$
- $\mathbf{W}_1$, $\mathbf{W}_2$: mxn matrices of r.v.’s
- $\mathbf{A}_1$, $\mathbf{A}_2$: kxm matrices of constants
- $\mathbf{B}$: nxl matrix of constants
$\mathbf{\mu} = E(\mathbf{X})$: mean of $\mathbf{X}$
$Cov(\mathbf{X})= E[(\mathbf{X}-\mathbf{\mu})(\mathbf{X}-\mathbf{\mu})’]= E[\mathbf{X}\mathbf{X}’] - \mathbf{\mu}\mathbf{\mu}’ = [\sigma_{ij}]$ : variance-covariance matrix (분산-공분산 행렬, nxn)
$Cov(\mathbf{AX}) = \mathbf{A}Cov(\mathbf{X})\mathbf{A}’$
$Cov(\mathbf{X})$ is p.s.d. i.e. $\mathbf{a}Cov(\mathbf{X})\mathbf{a}’ \geq 0$

$(X_1,\cdots,X_n) \rightarrow (Y_1,\cdots,Y_n)$ s.t. $y_1= u_1(x_1,\cdots,x_n)$, $\cdots$, $y_n= u_n(x_1,\cdots,x_n)$

one-to-one transformation
- $x_1= w_1(y_1,\cdots,y_n$, $\cdots$, $x_n= w_n(y_1,\cdots,y_n)$
- jpdf of $Y_1,\cdots,Y_n$: $g(y_1,\cdots,y_n)= \vert J\vert f(w_1(y_1,\cdots,y_n),\cdots,w_n(y_1,\cdots,y_n))$
many-to-one transformation
- $\bigcup_{i=1}^{k} A_i = S$ and $A_i \cap A_j = \emptyset$인 exhaustive sets $A_1,\cdots, A_n$
- $A_i \longrightarrow \mathscr{T}$ is 1-1

$g(y_1,\cdots,y_n)= \sum_{i=1}^{k} \vert J_i\vert f(w_{1i}(y_1,\cdots,y_n),\cdots,w_{ni}(y_1,\cdots,y_n))$

12 Jul 2024