Example. Random variables $\xi_{1}, \xi_{2}, \ldots$ are independent and distributed according to the Poisson law with parameter $a=1$. Let $S_{n}=$ $=\xi_{1}+\ldots+\xi_{n}, n=1,2, \ldots$ Find the probability $\mathrm{P}\left(2 S_{300}>S_{600}-30\right)$.
Solution. The random variable $S_{300}$ is approximately normally distributed as $N(300,300)$, since $n=300, \quad m=\mathrm{M} \xi_{1}=1$, $\sigma^{2}=\mathrm{D} \xi_{1}=1$. The random variable $S_{600}-S_{300}$ has exactly the same distribution, and the random variables $S_{300}$ and $S_{600}-S_{300}$ are independent. Let

$$
\eta=S_{300}, \quad \zeta=S_{600}-S_{300}
$$

Then $S_{600}=\eta+\zeta$ and the inequality $2 S_{300}>S_{600}-30$ can be written as $\zeta-\etaS_{600}-30\right)=\mathrm{P}(\zeta-\etaS_{600}-30\right) \approx 0.8897$.

Conditions of the Problem. The random variables $\xi_{1}, \xi_{2}, \ldots$ are independent and follow the same specified distribution. Let $S_{n}=\xi_{1}+\ldots+\xi_{n}, \quad n=1,2, \ldots$ Find the specified probability.

1. Exponential distribution with parameter $\lambda=1$, $\mathrm{P}\left(S_{100} \in(90,100)\right)$.
2. Normal distribution $N(1,1), \mathrm{P}\left(S_{900} \geqslant 850\right)$.
3. Poisson distribution with parameter $a=1, \mathrm{P}\left(S_{400}30\right)$
6. Exponential distribution with parameter $\lambda=1$, $\mathrm{P}\left(S_{1000}-S_{600}>390\right)$.
7. Normal distribution $N(1,1), \mathrm{P}\left(\left|S_{500}-S_{100}\right|S_{300}+5\right)$.
9. Binomial distribution with parameters $n=9, p=0.5$, $\mathrm{P}\left(S_{100}+S_{200}550\right)$

Answers. 1. 0.3414. 2. 0.9522. 3. 0.8413. 4. 0.9167. 5. 0.0455. 6. 0.6915. 7. 0.8413. 8. 0.1103. 9. 0.0680. 10. 0.9093.

## Chapter 7

## MATHEMATICAL STATISTICS

When studying the topic of MATHEMATICAL STATISTICS, you will learn to extract useful information from measurement results, survey data, and other factual material, as well as use this material to formulate and test hypotheses.

Problems 1-6 are devoted to using statistical data to find distribution laws. By representing statistical data in the form of clear graphs, we formulate a hypothesis about the distribution law of these data. Such a law usually contains one or several parameters, the numerical values of which need to be clarified.

Problems 7-12 are devoted to using statistical data to find numerical characteristics of distribution laws and, in particular, unknown parameters appearing in these laws.

You will learn to find point estimates of the expected value and variance, as well as apply the method of moments and the method of maximum likelihood to determine the parameters of the distribution.