Example 18. Independent random variables $X, Y, Z$ are normally distributed, with $M(X)=1, D(X)=2 ; M(Y)=-2, D(Y)=3$; $M(Z)=5, D(Z)=4$. Write the probability density function and the distribution function of their sum.
Solution. Since $M(X+Y+Z)=M(X)+M(Y)+M(Z)=$ $=1+(-2)+5=4, \quad D(X+Y+Z)=D(X)+D(Y)+D(Z)=2+3+4=9, \quad$ then the sum $X+Y+Z$ has a normal distribution with parameters $a=4$, $\sigma=3$. Therefore,

$$
p(x)=\frac{1}{3 \sqrt{2 \pi}} e^{-(x-4)^{2} / 18}, \quad F(x)=\frac{1}{2}+\Phi\left(\frac{x-4}{3}\right)
$$