721. Find explicit formulas for the distribution of a continuous two-dimensional random variable ( $X, Y$ ), given by the probability density function $f(x, y)=6 y$ in the region bounded by the lines $y=0, y=x, x=1$.
Solution. Let's find the probability density of the component $\mathrm{X}$:

$$
f_{1}(x)=\int_{0}^{x} f(x, y) d y=6 \int_{0}^{x} y \mathrm{~d} y=3 x^{2}(0<x<1)
$$

Simulate $X$ according to rule 2 (3):

$$
3 \int_{0}^{x_{i}} x^{2} \mathrm{~d} x=r_{i}
$$

From this, we obtain the explicit formula for calculating possible values of $X$:

$$
x_{i}=\sqrt[3]{r_{i}}
$$

Find the conditional probability density of the component $Y$:

$$
\psi(y \mid x)=f(x, y) / f_{i}(x)=(6 y) /\left(3 x^{2}\right)=(2 y) / x^{2} .
$$

Simulate $Y$ according to rule 2 (3):

$$
\left(2 / x^{2}\right) \int_{0}^{y_{i}} y \mathrm{~d} y=r_{i}^{\prime}
$$

From this, we obtain the explicit formula for calculating possible values of $Y$:

$$
y_{i}=x_{i} \sqrt{r_{i}^{\prime}}
$$

where $x_{i}$ is found using formula (*).