318. The diameter of a circle $x$ is measured approximately, with $a \leqslant x \leqslant b$. Considering the diameter as a random variable $X$, uniformly distributed in the interval $(a, b)$, find the mathematical expectation and variance of the area of the circle.
Solution. 1. Find the expected value of the area of the circle - a random variable $Y=\varphi(K)=\pi X^{2} / 4$ - using the formula

$$
M[\varphi(X)]=\int_{a}^{b} \varphi(x) f(x) d x
$$

Substituting $\varphi(x)=\pi x^{2} / 4, f(x)=1 /(b-a)$ and performing the integration, we get

$$
M\left[\pi X^{2} / 4\right]=\pi\left(b^{2}+a b+a^{2}\right) / 12
$$