Example 10. The random variable $X$ is distributed according to the normal law, with $M(X)=10, \quad D(X)=4$. Find:

$$
\text { 1) } P(12<X<14) ; 2) P(8<X<12)
$$
Solution. From the condition, it follows that $a=10, \sigma(x)=\sqrt{D(X)}=2$. According to formula (3.5.2), we find:

$$
\begin{gathered}
P(12<X<14)=\Phi\left(\frac{14-10}{2}\right)-\Phi\left(\frac{12-10}{2}\right)=\Phi(2)-\Phi(1)= \\
=0.477250-0.341345=0.135905 \\
P(8<X<12)=\Phi\left(\frac{12-10}{2}\right)-\Phi\left(\frac{8-10}{2}\right)=\Phi(1)-\Phi(-1)=2 \Phi(1)= \\
=2 \cdot 0.341345=0.682690
\end{gathered}
$$

Remark. Here, the corresponding values of the Laplace function (see appendix) and the oddness of this function are used.