Example 9. Write the probability density function and the distribution function of the normal random variable $X$, if $M(X)=3, \quad D(X)=4$.
Solution. From the condition, it follows that $a=M(X)=3, \sigma(x)=\sqrt{D(X)}=$ $=2$. Substituting these values into formula (3.5.1), we find the density function of the given random variable $X$:

$$
p(x)=\frac{1}{2 \sqrt{2 \pi}} e^{-(x-3)^{2} / 2 \cdot 2^{2}}=\frac{1}{2 \sqrt{2 \pi}} e^{-(x-3)^{2} / 8}
$$

According to formula (3.5.15), we write the distribution function of this random variable $X$.

$$
F(x)=\frac{1}{2}+\Phi\left(\frac{x-3}{2}\right)
$$