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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

# 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