Example 2. The random variable $X$ is uniformly distributed on the interval $[\alpha, \beta]$. Find the probability of its values falling into the interval $(\gamma, \delta)$, which belongs to the interval $[\alpha, \beta]$.
Solution. Using formula (2.3.3), we find

$$
\begin{aligned}
P(\gamma<X<\delta)= & \int_{\gamma}^{\delta} p(x) d x=\int_{\gamma}^{\delta} \frac{1}{\beta-\alpha} d x=\frac{1}{\beta-\alpha} \int_{\gamma}^{\delta} d x=\frac{\delta-\gamma}{\beta-\alpha} \\
& P(\gamma<X<\delta)=\frac{\delta-\gamma}{\beta-\alpha}
\end{aligned}
$$

Since $\delta-\gamma$ is the length of the interval $(\gamma, \delta)$, and $\beta-\alpha$ is the length of the segment $[\alpha, \beta]$, formula (3.4.4) expresses the probability of a point landing in the interval $(\gamma, \delta)$ when thrown onto the segment $[\alpha, \beta]$, i.e., the geometric definition of probability (see formula (1.5.3)).

Remark 1. The phrase "choose a point $x$ at random on the segment $[\alpha, \beta]$" means that the coordinate of the point $x$ represents a random variable with a uniform distribution of probabilities on this segment.

Example 3. Find the value of $C$ in formula (3.4.2), defining the uniform distribution of a two-dimensional random variable $(X, Y)$ in the region $G$.

Solution. The probability of the point $(X, Y)$ landing in any region $g$ lying within the region $G$ is proportional to the area $S_{g}$ of the region $g$:

$$
P((X, Y) \in g)=C \cdot S_{g}
$$

Since landing in the region $G$ is a certain event, then

$$
P((X, Y) \in G)=C \cdot S_{G}=1
$$

from which $C=1 / S_{G}$. Substituting this value into the first formula, we get

$$
P((X, Y) \in g)=\frac{S_{g}}{S_{G}}
$$

Comparing this formula with formula (1.5.1), we conclude that the geometric definition of probability has been obtained.

Remark 2. A two-dimensional random variable $(X, Y)$, where $X, Y$ are the coordinates of a point randomly thrown into the region $G$, is uniformly distributed in this region.