[수리통계학 I] Multivariate Distributions

6, 7강 요약 정리

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

6강 링크
7강 링크

목차
1. Distributions of Two Random Variables
2. Examples
3. Transformations: Bivariate r.v.s
4. Examples- transformation

Distributions of Two Random Variables

1. $(X_1, X_2)$: random vector
   • $\mathscr{D}= \lbrace(x_1, x_2): x_1= X_1(c), x_2= X_2(c), c\in\mathscr{C}\rbrace$
   • vector notation $\mathbf{X} = {X_1\choose X_2} = (X_1, X_2)’ = (X_1, X_2)^T$

• cdf of $\mathbf{X}$
  • $F_{X_1,X_2}(x_1, x_2) = P(X_1 \leq x_1, X_2 \leq x_2)$
  • $P(a_1 < X_1 \leq b_1, a_2 < X_2 \leq b_2)$
    $= F_{X_1,X_2}(b_1, b_2) - F_{X_1,X_2}(a_1, b_2) - F_{X_1,X_2}(b_1, a_2) + F_{X_1,X_2}(a_1, a_2)$
  • 
• joint prob. mass function if $\mathbf{X}$ is discrete: $p_{X_1,X_2}(x_1, x_2) = P(X_1= x_1, X_2= x_2)$
• joint pdf for continuous r.v.: $f_{X_1,X_2}(x_1, x_2) = \frac{\partial^2 F_{X_1,X_2}(x_1, x_2)}{\partial x_1 \partial x_2}$
  • $F_{X_1,X_2}(x_1, x_2) = \int_{-\infty}^{x_2} \int_{-\infty}^{x_1} f_{X_1,X_2}(w_1, w_2) \, dw_1 dw_2$
• Marginal of $X_1$
  • cdf: $F_{X_1}(x_1)= \lim_{x_2 \to \infty} F_{X_1,X_2}(x_1, x_2)$
  • pmf: $p_{X_1}(x_1)= \sum_{x_2} p_{X_1,X_2}(x_1, x_2)$
  • pdf: $f_{X_1}(x_1)= \int_{-\infty}^{\infty} f_{X_1,X_2}(x_1, x_2) \, dx_2$

  -

• continuous: $E[g(X_1, X_2)]= \iint g(x_1, x_2)f(x_1, x_2) \, dx_1 dx_2$
  • $\iint \vert g(x_1, x_2)\vert f(x_1, x_2) \, dx_1 dx_2 < \infty$
• discrete: $\sum_{x_1} \sum_{x_2} g(x_1, x_2)p(x_1, x_2)$
  • $\sum_{x_1} \sum_{x_2} \vert g(x_1, x_2)\vert p(x_1, x_2) < \infty$
• Linearity property of expectation: $E[k_1g_1(X_1, X_2) + k_2g_2(X_1, X_2)] = k_1E[g_1(X_1, X_2)] + k_2E[g_2(X_1, X_2)]$

-

• $M_{\mathbf{X}}(\mathbf{t}) = E[e^{\mathbf{t}’\mathbf{X}}]$, $\mathbf{t}= (t_1, t_2)^T$, $||\mathbf{t}|| < h$, $h>0$
  $= M_{X_1, X_2}(t_1, t_2)= E[e^{t_1X_1 + t_2X_2}]$ : mgf of $\mathbf{X}$

• $M_{X_1, X_2}(t_1, 0) = \iint e^{t_1x_1}f(x_1, x_2) \, dx_1dx_2 = \int e^{t_1x_1} \lbrace\int f(x_1, x_2) \, dx_2\rbrace \, dx_1$
  $= \int e^{t_1x_1} f_{X_1}(x_1) \, dx_1 = E[e^{t_!x_!}] = M_{X_1}(t_1)$ : marginal mgf of $X_1$

  • similarly, $M_{X_1, X_2}(0, t_2)= M_{X_2}(t_2)$: marginal mgf of $X_2$

Example 1

$f_{X_1,X_2}(x_1, x_2)= x_1 + x_2$, $0 < x_1 < 1$, $0 < x_2 < 1$이다.
$P(X_1 \leq \frac{1}{2})$와 $P(X_1 + X_2 \leq 1)$을 구해보자.

1. $P(X_1 \leq \frac{1}{2})$
   • $P(X_1 \leq \frac{1}{2})= P(X_1 \leq \frac{1}{2}, 0 \leq x_2 \leq 1)$
     $= \int_{0}^{\frac{1}{2}} f_1(x_1) \, dx_1$

   • $f_{X_1}(x_1) = \int f_{X_1,X_2}(x_1, x_2) \, dx_2$
     $= \int_{0}^{1} (x_1 + x_2) \, dx_2 = x_1 + \frac{1}{2}$

   • $\therefore P(X_1 \leq \frac{1}{2}) = \int_{0}^{\frac{1}{2}} (x_1 + \frac{1}{2}) \, dx_1= \frac{3}{8}$
2. $P(X_1 + X_2 \leq 1)$
   • $P(X_1 + X_2 \leq 1)= \int_{0}^{1} \int_{0}^{1-x_2} f_{X_1,X_2}(w_1, w_2) \, dw_1 dw_2$
     $= \int_{0}^{1} \int_{0}^{1-x_2} (x_1+x_2) \, dw_1 dw_2 = \frac{1}{3}$

Example 2

$f(x_1, x_2)= 8x_1x_2I(0 < x_1 < x_2 < 1)$일 때, $E(X_1{X_2}^2)$와 $E(X_2)$ 그리고 $E[7X_1{X_2}^2 + 5X_2]$를 구해보자.

1. $E(X_1{X_2}^2) = \int_{0}^{1} \int{0}^{x_2} x_1{x_2}^2 8x_1 x_2 \, dx_1 dx_2 = \frac{8}{21}$
2. $E(X_2)= \int_{0}^{1} x_2 f_{x_2}(x_2) \, dx_2 = \int_{0}^{1} x_2 [\int_{0}^{x_2} 8x_1 x_2 \, dx_1] \, dx_2 = \frac{4}{5}$
3. $E[7X_1{X_2}^2 + 5X_2]= \frac{20}{3}$

Example 3

$f(x,y)= e^{-y} I(0 < x < y < \infty)$: jpdf(joint pdf) of $(X, Y)$일 때 mgf를 구해보자.

• $M(t_1, t_2)= \int_{0}^{\infty} \int_{x}^{\infty} e^{t_1x+t_2y}e^{-y} \, dydx = \frac{1}{(1-t_1-t_2)(1-t_2)}$
• mgf of X: $M(t_1,0) = \frac{1}{1-t_1}$
• mgf of Y: $M(0, t_2)= \frac{1}{(1-t_2)^2}$

적분 구간 신경 쓰기!

Transformations

in Bivariate R.V.s

• goal: $f_{X_1, X_2}(x_1, x_2)$가 주어졌을 때, $Y= g(x_1, x_2)$의 PDF를 구하자!

• $X_1$과 $X_2$의 jpdf를 알고 있을 때, $Y= g(X_1, X_2)$의 분포를 구하는 방법에는 2가지가 있다.

  1. Find cdf of Y → take derivative
  2. Use transformation technique

1. discrete case

• transformation on $Y= g(x_1, x_2)$

• $(X_1, X_2)$: discrete r.v. with jpdf $p_{X_1, X_2}(x_1, x_2)$ and support $S$

  • 

  • $p_{Y_1, Y_2}(y_1, y_2) = p_{X_1, X_2}(w_1(y_1,y_2), w_2(y_1,y_2))$, $(y_1, y_2) \in \mathscr{I}$
  • $\Rightarrow$ pmf of $Y_i$: $P_{Y_1}(y_1) = \sum_{y_2} P_{Y_1, Y2}(y_1, y_2)$

2. Continuous case

1. cdf technique: $F_Y(y)= p(g(x_1, x_2) \leq y) \rightarrow f_Y(y)= F_Y’(y)$
2. transformation
   + $X_1$, $X_2$의 jpdf $f_{X_1, X_2}(x_1, x_2)$와 support $S$
   + 기본 변환은 discrete case와 똑같이 생각한다. $(x_1, x_2) \rightarrow (y_1, y_2)$일 때, $y_1= u_1(x_1, x_2), y_2= u_2(x_1, x_2)$이고 역함수는 $x_1= w_1(y_1, y_2), x_2= w_2(y_1, y_2)$이다.
   + Jacobian $J = \begin{pmatrix} \frac{\partial x_1}{\partial y_1} & \frac{\partial x_1}{\partial y_2} \\
   \frac{\partial x_2}{\partial y_1} & \frac{\partial x_2}{\partial y_2} \end{pmatrix}$
   + jpdf of $Y_1$ and $Y_2$: $f_{Y_1, Y_2}(y_1, y_2)= f_{X_1, X_2}(w_1(y_1,y_2), w_2(y_1,y_2)) \vert J \vert$, $(y_1, y_2) \in \mathscr{T}$

Example 1: discrete

$p_{X_1, X_2}(x_1, x_2)= \frac{u_1^{x_1}u_2^{x_2}e^{-u_1-u_2}}{x_1!x_2!}$, $x_1= 0,1,2,\cdots$, $x_2=0,1,2,\cdots$일 때, $Y_1= X_1 + X_2$의 pdf를 구하여라.

• 이항정리: $(a+b)^n= \sum_{k=0}^{n} {n\choose k}a^k b^{n-k}$

Example 2: continuous

$f_{X_1, X_2}(x_1, x_2)= I(0<x_1<1, 0<x_2<1)$일 때, $Y_1= X_1 + X_2$의 pdf를 구하여라.

1. cdf technique
   • 
2. transformation
   • 

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