bivariate case 라고 가정

Discrete case

Let $Y_1 = g_1(x_1,x_2) \space and \space Y_2 = g_2(x_1,x_2)$, then
$g^{-1}(y_1,y_2) = \left\{ (x_1,x_2) \in support (x) : y_1 = g_1(x_1,x_2), y_2 = g_2(x_1,x_2) \right\}$

$P_{Y_1,Y_2}(y_1,y_2) = P(Y_1 = y_1, Y_2 = y_2) = P(g_1(x_1,x_2) = y_1, g_2(x_1,x_2) = y_2) = P((x_1,x_2) \in g^{-1}(y_1,y_2)) = \sum_{ (x_1,x_2) \in g^{-1}(y_1,y_2) } P_{X_1,X_2} (x_1,x_2)$

ex1)
$X_1 \sim Poisson(\lambda_1), X_2 \sim Poisson(\lambda_2), X_1, X_2: indep.$
$P_{X_1,X_2} (x_1,x_2) = \frac{ \lambda_1^{x_1} \lambda_2^{x_2} e^{-\lambda_1} e^{-\lambda_2}}{ x_1! x_2! } \space (x_1=0,1,2..., \space x_2=0,1,2,...)$

Let $Y_1=X_1+X_2, Y_2=X_2$
=> $X_2 = Y_2, X_1 = Y_1 - Y_2(Y_1 \geq Y_2)$
$g^{-1}(y_1,y_2) = \left\{ (x_1,x_2) \in support (x) : x_1 = y_1 - y_2, x_2 = y_2 \right\} = (y_1-y_2,y_2)$

$P_{Y_1,Y_2} (y_1,y_2)=\sum_{(x_1,x_2) \in g^{-1}(y_1,y_2)} P_{X_1,X_2} (x_1,x_2) = \sum_{(x_1,x_2) \in (y_1-y_2,y_2)} P_{X_1,X_2} (x_1,x_2) = P_{X_1,X_2} (y_1-y_2,y_2) = \frac{ \lambda_1^{y_1-y_2} \lambda_2^{y_2} e^{-\lambda_1 - \lambda_2} }{ (y_1-y_2)! y_2!} (y_1=y_2,y_2 + 1,...$ and $y_2=0,1,2,...,y_1)$

• marginal pmf of $Y_1$
  
  
• marginal pmf of $Y_2$
  
  
• mgf of $Y_1$
  

ex1)
If $X_i \sim Poisson(\lambda_i), i=1,2,...,m$ & $X_i$'s are indep., then
$Y=X_1+...+X_m \sim Poisson(\sum_{i=1}^m \lambda_i)$
pf)

ex2)
If $X_i \sim B(n_i,p) ,i=1,...,m$ & $X_i$'s are indep., then
$Y=X_1+...+X_m \sim B(\sum_{i=1}^{n} n_i, p)$
pf)

Continuous case

If $\underline{h}$ is an one-to-one function that maps $support(x)$ onto $support(y)$,
$f_{Y_1,Y_2} (y_1,y_2) = |J|f_{X_1,X_2} (h_1(y_1,y_2), h_2(y_1,y_2))$
where $x_1=h_1(y_1,y_2)$ & $x_2=h_2(y_1,y_2)$ & $|J|=\begin{vmatrix} det \begin{bmatrix} \frac{ \partial h_1}{ \partial y_1} & \frac{ \partial h_1}{ \partial y_2} \\ \frac{ \partial h_2}{ \partial y_1} & \frac{ \partial h_2}{ \partial y_2} \\ \end{bmatrix} \end{vmatrix}$

ex1)
$f(x_1,x_2) = 1,0<x_1<1,0<x_2<1$

Let $Y_1=X_1+X_2,Y_2=X_1-X_2$
=> $X_1=\frac{Y_1+Y_2}{2}, X_2=\frac{Y_1-Y_2}{2}$
=> $|J|$ = $\begin{vmatrix} det \begin{bmatrix} 1/2 & 1/2 \\ 1/2 & -1/2 \end{bmatrix} \end{vmatrix}=1/2$
$f_{Y_1,Y_2} (y_1,y_2) = 1/2$

• support of $(Y_1,Y_2)$
  
  
• marginal pdf
  $f_{Y_1} (y_1) = \left\{\begin{matrix} \int_{-y_1}^{y_1} \frac{1}{2} dy_2 = y_1 \space (0 < y_1 \leq 1) \\ \int_{y_1-2}^{2-y_1} \frac{1}{2} dy_2 = 2-y_1 \space (1 < y_1 \leq 2) \end{matrix}\right.$
  
  $f_{Y_2} (y_2) = \left\{\begin{matrix} \int_{-y_2}^{y_2+2} \frac{1}{2} dy_1 = y_2+1 \space (-1 < y_2 \leq 0) \\ \int_{y_2}^{2-y_2} \frac{1}{2} dy_1 = 1-y_2 \space (0 < y_2 < 1) \end{matrix}\right.$

ex2) Beta Distribution
$X_1 \sim Gamma(\alpha, \lambda),X_2 \sim Gamma(\beta, \lambda)$ & $X_1, X_2: indep.$
Let $Y_1=\frac{X_1}{X_1+X_2}, Y_2=X_1+X_2$

• Joint pdf of $Y_1,Y_2$
  $X_1=Y_1 Y_2, \space X_2=Y_2 - Y_1 Y_2 =Y_2(1-Y_1)$
  $|J|=\begin{vmatrix} det \begin{bmatrix} y_2 & y_1 \\ -y_2 & 1-y_1 \end{bmatrix} \end{vmatrix} = y_2$
  
  $f_{X_1,X_2} (x_1,x_2) = \frac{1}{ \Gamma(\alpha) \lambda^{\alpha}} x_1^{\alpha - 1} e^{-\frac{x_1}{\lambda}}\frac{1}{ \Gamma(\beta) \lambda^{\beta}} x_2^{\beta - 1} e^{-\frac{x_2}{\lambda}} = \frac{1}{ \Gamma(\alpha) \Gamma(\beta) \lambda^{\alpha + \beta}} x_1^{\alpha - 1} x_2^{\beta - 1}e^{-\frac{x_1 + x_2}{\lambda}}$
  
  $f_{Y_1,Y_2} (y_1,y_2) = y_2 f_{X_1,X_2} (y_1y_2, y_2(1-y_1)) = y_2\frac{1}{ \Gamma(\alpha) \Gamma(\beta) \lambda^{\alpha + \beta}} (y_1y_2)^{\alpha - 1} (y_2(1-y_1))^{\beta - 1}e^{-\frac{y_2}{\lambda}} = \frac{1}{ \Gamma(\alpha) \Gamma(\beta) \lambda^{\alpha + \beta}} y_1^{\alpha - 1} y_2^{\alpha + \beta -1} (1-y_1)^{\beta - 1} e^{-\frac{y_2}{\lambda}}$
  
  
• support of $Y_1,Y_2$
  
  
• marginal pdf
  $f_{Y_1} (y_1) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)}y_1^{\alpha-1}(1-y_1)^{\beta-1},0 < y_1 < 1 \sim Beta(\alpha,\beta)$
  pf)
  
  $f_{Y_2} (y_2) = \frac{1}{\Gamma(\alpha + \beta) \lambda^{\alpha + \beta}} y_2^{\alpha + \beta - 1} e^{-\frac{y_2}{\lambda}},0 < y_2 < \infty \sim Gamma(\alpha + \beta, \lambda)$
  pf)
  

✔︎ beta function $B(\alpha, \beta)$

$B(\alpha, \beta) = \int_{0}^{1} x^{\alpha - 1} (1-x)^{\beta - 1} dx = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)}$

✔︎ Expectation of beta distribution

$Y_1 \sim Beta(\alpha,\beta)$
$E(Y_1) = \frac{\alpha}{\alpha + \beta}$
$Var(Y_1) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$