Example 6.30. A random variable $X$ is normally distributed. The mathematical expectation and standard deviation of this variable are 2 and 5, respectively. Find the probability that in the experiment, the RV $X$ will take a value belonging to the interval $(1 ; 4)$.
Solution. We will use the formula:

$$
P(\alpha<X<\beta)=\Phi \frac{\beta-a}{\sigma}-\Phi \frac{\alpha-a}{\sigma}
$$

According to the condition: $\alpha=1, \beta=4, a=2, \sigma=5$, therefore,

$$
P(1<X<4)=\Phi \frac{4-2}{5}-\Phi \frac{1-2}{5}=\Phi(0.4)-\Phi(-0.2)
$$

Since the Laplace function is odd, then

$$
\Phi(-0.2)=-\Phi(0.2)
$$

Thus,

$$
P(1<X<4)=\Phi(0.4)+\Phi(0.2)
$$

From the table we find:

$$
\Phi(0.4)=0.1554 ; \quad \Phi(0.2)=0.0793 .
$$

The required probability is:

$$
P(1<X<4)=0.1554+0.0793=0.2347 .
$$