5. Find the explicit formulas for the distribution of the continuous two-dimensional random variable ( $X, Y$ ), given by the probability density function $f(x, y)=(3 / 4) x y^{2}$ in the region bounded by the lines $x=0, y=0, x=1, y=2$.
Solution. Components $X$ and $Y$ are independent, as the joint probability density $f(x, y)=(3 / 4) x y^{2}$ can be represented as the product of two functions, one depending only on $x$, and the other only on $y$.

Let's find the density of the distribution of component $X$:

$$
f_{1}(x)=\int_{0}^{2} f(x, y) \mathrm{d} y=(3 / 4) x \int_{0}^{2} y^{2} \mathrm{~d} y=2 x
$$

Thus, $f_{1}(x)=2 x(0<x<1)$.

Let's simulate $X$ according to rule 2 ( $\$ 3$ ):

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

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

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

Let's find the density of the distribution of component $Y$:

$$
f_{2}(y)=\int_{0}^{1} f(x, y) \mathrm{d} x=(3 / s) y^{2} \int_{0}^{1} x \mathrm{~d} x=(3 / 8) y^{2}
$$

Thus, $f_{3}(y)=(3 / 8) y^{2}(0<y<2)$.

Let's simulate $Y$ according to rule 2 (§ 3 ):

$$
(3 / 8) \int_{0}^{\eta_{i}} y^{2} \mathrm{~d} y=r_{i}{ }^{i}
$$

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

$$
y_{i}=2 \sqrt[3]{r_{i}^{i}}
$$