Example 22. The root mean square deviation of a random variable distributed according to the normal law is $2 \mathrm{~cm}$, and the mathematical expectation is 16 cm. Find the limits within which with a probability of 0.95 the value of the random variable should be expected.
Solution. According to formula (3.5.4) when $\sigma=2$, we have $2 \Phi\left(\frac{\delta}{\sigma}\right)=0.95$ or $\Phi\left(\frac{\delta}{\sigma}\right)=0.475$. From the table of values of the Laplace function, we find that $\delta / 2=1.96$. Therefore, $\delta / 2=1.96$, from which $\delta=3.92$.

Thus, with a probability of 0.95, we can expect that the deviation of the random variable from the expected value will not exceed 3.92 cm in absolute value, i.e., $|X-16| \leq 3.92$, from which $12.08 \leq X \leq 19.92$. Therefore, all values of the normal random variable will be in the interval (12.08; 19.92).