Example 6. Find the mathematical expectation of a random variable $X$, having an exponential distribution.
Solution. We will use formula (2.4.8) and formula (3.6.2):

$$
M(X)=\int_{-\infty}^{+\infty} x p(x) d x=\int_{-\infty}^{0} x p(x) d x+\int_{0}^{+\infty} x p(x) d x=\int_{-\infty}^{0} 0 \cdot x d x+\int_{0}^{+\infty} x \alpha e^{-\alpha x} d x
$$

$$
M(X)=\alpha \int_{0}^{+\infty} x e^{-\alpha x} d x
$$

Applying the integration by parts formula

$$
\int_{0}^{+\infty} u d v=\left.u v\right|_{0} ^{+\infty}-\int_{0}^{+\infty} v d u
$$

assuming $u=x, d v=\alpha e^{-\alpha x} d x=-d\left(e^{-\alpha x}\right)$, we find

$$
\begin{aligned}
& \alpha \int_{0}^{+\infty} x e^{-\alpha x} d x=-\int_{0}^{+\infty} x d\left(e^{-\alpha x}\right)=-\left.x e^{-\alpha x}\right|_{0} ^{+\infty}+\int_{0}^{+\infty} e^{-\alpha x} d x= \\
& =0-\left.\frac{1}{\alpha} e^{-\alpha x}\right|_{0} ^{+\infty}=-\frac{1}{\alpha}\left(e^{-\infty}-e^{0}\right)=\frac{1}{\alpha} . \quad M(X)=\frac{1}{\alpha}
\end{aligned}
$$

Thus, the expected value of a random variable distributed according to the exponential law is equal to the reciprocal of the parameter $\alpha$.

Example 7. Find the variance and standard deviation of a random variable that has an exponential distribution.

**Solution.** We will find the variance using formula (2.5.14)

$$
D(x)=\int_{0}^{+\infty} x^{2} p(x) d x-(M(x))^{2}
$$

Using formula (3.6.2) and the results from Example 6, we rewrite the expression for the variance as:

$$
D(x)=\int_{0}^{+\infty} x^{2} \alpha e^{-\alpha x} d x-1 / \alpha^{2}
$$

Integrating by parts twice, we get:

$$
\int_{0}^{+\infty} x^{2} \alpha e^{-\alpha x} d x=2 / \alpha^{2}
$$

Therefore,

$$
D(x)=\alpha \int_{0}^{+\infty} x^{2} e^{-\alpha x} d x-1 / \alpha^{2}=\frac{2}{\alpha^{2}}-\frac{1}{\alpha^{2}}=\frac{1}{\alpha^{2}}
$$

$$
D(x)=\frac{1}{\alpha^{2}}
$$

To find the standard deviation, we take the square root of the variance:

$$
\sigma(x)=\sqrt{D(X)}=\frac{1}{\alpha}
$$

Remark. The expected value and the standard deviation of the exponential distribution are equal to each other (see Example 6).