118. The random variable $X$ is given by the distribution function:

$$
F(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x \leq -c \\
\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c} & \text { if } & -c < x \leq c \\
1 & \text { if } & x > c
\end{array}\right.
$$

( arcsine law ).

Find the mathematical expectation of this variable.
Solution. Let's find the probability density function of the random variable $X: f(x)=F^{\prime}(x)$. For $x \leq -c$ and for $x > c$, the function $f(x)=0$. For $-c < x \leq c$, we have

$$
f(x)=\left(\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c}\right)^{\prime}=\frac{1}{\pi \sqrt{c^{2}-x^{2}}}
$$

The expected value of the random variable $X$ will be found using formula (54) with $a=-c$ and $b=c$:

$$
M(X)=\int_{-c}^{c} \frac{x d x}{\pi \sqrt{c^{2}-x^{2}}}
$$

Considering that the integrand is an odd function and the limits of integration are symmetric with respect to the origin, we conclude that the integral is zero; therefore, $M(X)=0$.