## Example 5. Given the function

$$
f(x)= \begin{cases}0 & \text { if } x \leq 0 \\ c e^{-\alpha x} & \text { if } x>0 \quad(\alpha>0)\end{cases}
$$

For what value of the constant $c$ is the function $f(x)$ a probability density function of some random variable $X$?
Solution. First of all, it must be $c \geq 0$. To determine the value of $c$, we will use condition (2.3.6):

$$
\int_{-\infty}^{+\infty} f(x) d x=1, c \int_{0}^{+\infty} e^{-\alpha x} d x=1,-\left.\frac{c}{\alpha} e^{-\alpha x}\right|_{0} ^{+\infty}=1, \frac{c}{\alpha}=1, c=\alpha
$$

Therefore, the probability density function has the form

$$
p(x)= \begin{cases}0 & \text { if } x \leq 0 \\ \alpha e^{-\alpha x} & \text { if } x>0 ; \alpha>0\end{cases}
$$