133. The student remembers that the probability density function of the exponential distribution has the form $f(x)=0$ for $x<0$, and for $x \geq 0$

$f(x)=C e^{-\lambda x}$; however, he forgot the value of the constant $C$. Find the parameter $C$.
S o l u t i o n. To find $C$, we will use the property of the probability density function: $\int_{-\infty}^{\infty} f(x) d x=1$. For $x<0$, the function $f(x)=0$, and for $x \geq 0$, the function $f(x)=C e^{-\lambda x}$, so the condition must be satisfied: $\int_{0}^{\infty} C e^{-\lambda x} d x=1$.

We will find the improper integral with an infinite upper limit of integration on the left side of this equation:

$$
\int_{0}^{\infty} C e^{-\lambda x} d x=C \cdot \lim _{b \rightarrow \infty} \int_{0}^{b} e^{-\lambda x} d x=-\frac{C}{\lambda} \lim _{b \rightarrow \infty} e^{-\left.\lambda x\right|_{0} ^{b}}=-\frac{C}{\lambda} \lim _{b \rightarrow \infty}\left(e^{-\lambda b}-e^{0}\right)=\frac{C}{\lambda}
$$

From the equality $\frac{C}{\lambda}=1$, we get $C=\lambda$.