719. Find explicit formulas for simulating a continuous two-dimensional random variable $(X, Y)$, if the component $X$ is given by the probability density function $f_{1}(x) = x / 2$ in the interval $(0; 2)$; the component $Y$ is uniformly distributed in the interval $\left(x_{i}, x_{i}+3\right)$ with density $f_{2}(y)=1 / 3$, where $x_{i}$ is the simulated possible value of $X$.
Solution. We will generate the component $X$ according to rule 2 (§ 3):

$$
\int_{0}^{x_{l}}(x / 2) \mathrm{d} x=r_{i}
$$

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

$$
x_{i}=2 \sqrt{r_{i}}
$$

We will find the conditional distribution function of the component $\gamma$, taking into account that the value $Y$ is uniformly distributed in the interval $\left(x_{i}, x_{i}+3\right)$.

We use rule 1 (§ 3): $\left(y_{i}-x_{i}\right) / 3=r_{i}^{\prime}$, where $r_{i}^{\prime}$ is a random number.

Solving this equation for $y_{i}$, we obtain an explicit formula for calculating possible values of $y$:

$$
y_{i}=3 r_{i}^{\prime}+x_{i}
$$

where $x$ is found using formula (*).