Example 9. The function $f(x)=a e^{-|x|}$ is given. For what value of $a$ can it be considered as the probability density function of some random variable $X$?
Solution. If $f(x)$ is a probability density, then condition (2.3.6) must be satisfied. Therefore,

$$
\begin{aligned}
& \int_{-\infty}^{+\infty} a e^{-|x|} d x=a \int_{-\infty}^{0} e^{x} d x+a \int_{0}^{+\infty} e^{-x} d x=\left.a e^{x}\right|_{-\infty} ^{0}-\left.a e^{-x}\right|_{0} ^{+\infty}= \\
& =a(1-0)-a(0-1)=2 a=1, \quad \text { i.e. } a=1 / 2
\end{aligned}
$$

Thus, the function $p(x)=\frac{1}{2} e^{-|x|}$ is the probability density function of some random variable.