353. Find the mathematical expectation of the exponential distribution

$$
f(x)=\lambda \mathrm{e}^{-\lambda x} \quad(x \geqslant 0) ; \quad f(x)=0 \quad(x<0) .
$$
Solution. We use the formula

$$
M(X)=\int_{-\infty}^{\infty} x f(x) \mathrm{d} x
$$

Considering that $f(x)=0$ for $x<0$ and $f(x)=\lambda \mathrm{e}^{-\lambda x}$ for $x \geqslant 0$, we get

$$
M(X)=\lambda \int_{0}^{\infty} x \cdot \mathrm{e}^{-\lambda x} \mathrm{~d} x
$$

Integrating by parts using the formula

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

setting $u=x, \mathrm{~d} v=\mathrm{e}^{-\lambda x} \mathrm{~d} x$ and completing the calculations, we finally get $M(X)=1 / \lambda$.

Thus, the expected value of the exponential distribution is the reciprocal of the parameter $\lambda$.