Example 6. A random variable $X$ is distributed according to the Poisson law. Find $P(X=3)$, if $a=4$, as well as the mathematical expectation and variance of the variable $X$.
Solution. Using formula (3.3.1), we find

$$
P(X=3)=\frac{4^{3}}{3!} e^{-4}=\frac{32}{3} \cdot \frac{1}{e^{4}}=10.6667 \cdot 0.0183=0.1952
$$

According to formulas (3.3.5) and (3.3.6), we obtain

$$
M(X)=4, \quad D(X)=4
$$