258. The random variable $X$ is defined on the entire $O x$ axis by the distribution function $F(x)=1 / 2+(1 / \pi) \operatorname{arctg}(x / 2)$. Find the possible value $x_{1}$, satisfying the condition:
with probability $1 / 4$ the random variable $X$ will take a value greater than $x_{i}$ in the experiment.
Solution. Events $Xx_{1}$ are opposite, so $P\left(Xx_{1}\right)=1$. Therefore, $P\left(Xx_{1}\right)=1-1 / 4=3 / 4$. Since $P\left(X=x_{1}\right)=0$, then

$$
P\left(X<x_{1}\right)=P\left(X=x_{1}\right)+P\left(X<x_{1}\right)=P\left(X<x_{1}\right)=3 / 4
$$

By the definition of the distribution function,

$$
P\left(X<x_{1}\right)=F\left(x_{1}\right)=1 / 2+(1 / \pi) \operatorname{arctg}\left(x_{1} / 2\right) .
$$

Therefore,

$$
1 / 2+(1 / \pi) \operatorname{arctg}\left(x_{1} / 2\right)=(3 / 4), \text { or } \operatorname{arctg}\left(x_{1} / 2\right)=\pi / 4
$$

Hence $x_{1} / 2=1$, or $x_{1}=2$.