Joint distribution of functions of variables

Definition:

• Joint variables but only for continuous’s Distribution of a function of a random variable
• Let $X_1$ and $X_2$ be jointly continuous random variables with Joint probability density function $f_{X_1,X_2}$.
• Suppose: $Y_1=g_1(X_1,X_2),Y_2=g_2(X_1,X_2)$ for some functions $g_1$ and $g_2$ satisfying:
  1. The equations $y_1=g_1(x_1,x_2)$ and $y_2=g_2(x_1,x_2)$ can be uniquely solved for $x_1$ and $x_2$ in terms of $y_1$ and $y_2$
  2. $g_1$ and $g_2$ have continuous partial derivatives at all $(x_1,x_2)$ and are such that the Jacobian determinant $J(x_1,x_2)\ne 0$
     • Take the absolute of jacobian
• Then $Y_1$ and $Y_2$ are jointly continous with JPDF, $f_{Y_1{Y}_2}(y_1,y_2)=f_{X_1,X_2}(x_1,x_2)|J(x_1,x_2){|}^{-1}$
  • Replace $x_1,x_2$ with $y_1,y_2$
• For $n>2$:
  • $Y_1=g_1(X_1,...,X_n),...,Y_n=g_n(X_1,...,X_n)$
  • Conditions:
    • $g_i$ all have continuous partial derivative and have unique solutions for $x_i$
    • Jacobian determinant, $J(x_1,..,x_n)\ne 0$
    • Then $f_{Y_1,...,Y_n}=f_{X_1,...,X_2}(x_1,...,x_n)|J(x_1,...,x_n){|}^{-1}$

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