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Please consider the following probability density function of two variables. \begin{eqnarray*} f(x_1,x_2) &=& \begin{cases} 2(x_1+x_2) & \text{for} \,\, 0 <= x_1 <= x_2 <= 1\\ 0 &\text{ otherwise } \\ \end{cases} \end{eqnarray*}\begin{eqnarray*} f(x_1,x_2) &=& \begin{cases} 2(x_1+x_2) & \text{for} \,\, 0 \le x_1 \le x_2 \le 1\\ 0 &\text{ otherwise } \\ \end{cases} \end{eqnarray*} I would like to generate two random values $X_1$ and $X_2$ such that there distribution is consistent with the above probability density function.

If the above probability density function was of just one variable, then I would solve the above problem with the following steps.
1) I would finding find the distribution function $F$ by integrating.
2) I would generate a random number $a$ on the interval $[0,1]$ using the uniform distribution.
3) I would find a number $X$, such that $F(X) = a$. Then $X$ would be the random number I seek.
I am wondering, if whether a similar method would work for a function of two variables.
Bob

Please consider the following probability density function of two variables. \begin{eqnarray*} f(x_1,x_2) &=& \begin{cases} 2(x_1+x_2) & \text{for} \,\, 0 <= x_1 <= x_2 <= 1\\ 0 &\text{ otherwise } \\ \end{cases} \end{eqnarray*} I would like to generate two random values $X_1$ and $X_2$ such that there distribution is consistent with the above probability density function.

If the above probability density function was of just one variable, then I would solve the above problem with the following steps.
1) I would finding the distribution function $F$ by integrating.
2) I would generate a random number $a$ on the interval $[0,1]$ using the uniform distribution.
3) I would find a number $X$, such that $F(X) = a$. Then $X$ would be the random number I seek.
I am wondering, if a similar method would work for a function of two variables.
Bob

Please consider the following probability density function of two variables. \begin{eqnarray*} f(x_1,x_2) &=& \begin{cases} 2(x_1+x_2) & \text{for} \,\, 0 \le x_1 \le x_2 \le 1\\ 0 &\text{ otherwise } \\ \end{cases} \end{eqnarray*} I would like to generate two random values $X_1$ and $X_2$ such that there distribution is consistent with the above probability density function.

If the above probability density function was of just one variable, then I would solve the above problem with the following steps.
1) I would find the distribution function $F$ by integrating.
2) I would generate a random number $a$ on the interval $[0,1]$ using the uniform distribution.
3) I would find a number $X$, such that $F(X) = a$. Then $X$ would be the random number I seek.
I am wondering whether a similar method would work for a function of two variables.

1
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Generating Pairs of Random Numbers

Please consider the following probability density function of two variables. \begin{eqnarray*} f(x_1,x_2) &=& \begin{cases} 2(x_1+x_2) & \text{for} \,\, 0 <= x_1 <= x_2 <= 1\\ 0 &\text{ otherwise } \\ \end{cases} \end{eqnarray*} I would like to generate two random values $X_1$ and $X_2$ such that there distribution is consistent with the above probability density function.

If the above probability density function was of just one variable, then I would solve the above problem with the following steps.
1) I would finding the distribution function $F$ by integrating.
2) I would generate a random number $a$ on the interval $[0,1]$ using the uniform distribution.
3) I would find a number $X$, such that $F(X) = a$. Then $X$ would be the random number I seek.
I am wondering, if a similar method would work for a function of two variables.
Bob