Example 8. Find the probability of falling into the interval $(a, b)$ of values of a random variable $X$, distributed according to the exponential law.
Solution. We will use the distribution function (3.6.7) and formula $(2.2 .10)$ :

$$
P(a0$ and $F(x)=0$ for $x \leq 0, F(b)=1-e^{-\alpha b}$, $F(a)=1-e^{-\alpha a}$, ro

$$
\begin{gathered}
P(a<X<b)=F(b)-F(a)=\left(1-e^{-\alpha b}\right)-\left(1-e^{-\alpha a}\right) \\
P(a<X<b)=e^{-\alpha a}-e^{-\alpha b}
\end{gathered}
$$