Problem 5. In an urn, there are 9 balls, of which 5 are white and 4 are black. Four balls are randomly drawn from the urn. Let $X$ be the number of white balls among the drawn ones. Construct the distribution law of the discrete random variable $X$.

 untranslated text:
Задача 5. В урне находятся 9 шаров, из них 5 белых и 4 черных. Из урны наугад вынимаются 4 шара. Пусть $X$ - число белых шаров среди вынутых. Построить закон распределения ДСВ $X$.

 translated text:
Problem 5. In an urn, there are 9 balls, of which 5 are white and 4 are black. Four balls are randomly drawn from the urn. Let $X$ be the number of white balls among the drawn ones. Construct the distribution law of the discrete random variable $X$.
Solution. A similar problem in a more general form is considered in Chapter 2, Section VI (see problem 6): $N=9, M=5, n=4, k=0,1,2,3,4$.

Since $C_{9}^{4}=\frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1}=126$, the denominators of all 5 fractions are equal to 126. The results will be rounded to three decimal places:

$$
\begin{gathered}
p(X=0)=\frac{C_{5}^{0} \cdot C_{4}^{4}}{126}=\frac{1 \cdot 1}{126} \approx 0.008 ; \quad p(X=1)=\frac{C_{5}^{1} \cdot C_{4}^{3}}{126} \approx 0.159 \\
p(X=2)=\frac{C_{5}^{2} \cdot C_{4}^{2}}{126} \approx 0.476 ; \quad p(X=3)=\frac{C_{5}^{3} \cdot C_{4}^{1}}{126} \approx 0.317 \\
p(X=4)=\frac{C_{5}^{4} \cdot C_{4}^{0}}{126} \approx 0.04
\end{gathered}
$$

The distribution law is as follows:

| $x$ | 0 | 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $p$ | 0.008 | 0.159 | 0.476 | 0.317 | 0.040 |

## Exercises

1. Formulate the binomial distribution law with parameters $n=4$ and $p=0.65$.
2. In a batch of 6 items, 4 are standard. Three items are randomly selected. Formulate the distribution law for the number of standard items among the selected ones.
3. Write the first 4 terms of the Poisson distribution with parameter $\lambda=4$.
4. A device consists of three blocks operating independently of each other. The probability that each block will not fail during time $T$ is 0.9. Formulate the distribution law for the number of blocks that will fail during time $T$.
5. Formulate the geometric distribution law with parameter $p=0.5$.

## Answers

1. 0.0150; 0.1114; 0.3106; 0.3845; 0.1785. 2. 0; 0.2; 0.6; 0.2. 3. 0.018; 0.073; 0.147; 0.195. 4. 0.729; 0.243; 0.027; 0.001. 5. $\frac{1}{2}$; $\frac{1}{4}$; $\frac{1}{8}$; $\frac{1}{16}$.

## § 2. Numerical Characteristics of Discrete Random Variables

1$^{\circ}$. The mathematical expectation of a discrete random variable (DRV) $X$, taking values $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ with probabilities $p_{1}, p_{2}, p_{3}, \ldots, p_{n}$ respectively, is defined as

$$
M(X)=\sum_{\imath=1}^{n} p_{\imath} x_{\imath}=p_{1} x_{1}+p_{2} x_{2}+\ldots+p_{n} x_{n}
$$

If $X$ takes an infinite set of values, then instead of $n$ above the summation sign, $\infty$ is written, and on the right side of this equation, we have an infinite series that may converge or diverge.

$M(X)$ is the probabilistic average of the values of $X$ and represents the "center" of the DRV $X$. If we consider that at points $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ are concentrated masses $p_{1}, p_{2}, p_{3}, \ldots, p_{n}$ respectively, then $M(X)$ determines the center of mass (center of gravity) of this system of material points. The values of $X$ are distributed around $M(X)$.

$2^{\circ}$. The mathematical expectation $M(X)$ has the following properties:

1) if $C$ is a constant, then $M(C)=C$;
2) $M(C X)=C \cdot M(X)$
3) $M(X+Y)=M(X)+M(Y)$
4) if $X$ and $Y$ are independent DRVs, then $M(X \cdot Y)=M(X) \cdot M(Y)$.

$3^{\circ}$. The mathematical expectation of a random variable distributed according to a known law is expressed through the parameters of this law.

If the random variable $X$ is distributed according to the law:

- Bernoulli with parameters $p$ and $n$, then $M(X)=p n$;
- Poisson with parameter $\lambda$, then $M(X)=\lambda$;
- geometric with parameter $p$, then $M(X)=1 / p$;
- hypergeometric with parameters $N, M, n$, then $M(X)=\frac{n M}{N}$.

$4^{\circ}$. If $M(X)$ is the mathematical expectation of $X$, then the quantity $X-M(X)$ is called the deviation of $X$ from its mathematical expectation. The difference $X-M(X)$ is a random variable that can take both positive and negative values. It is easy to see that $M[X-M(X)]=0$.

The variance of a DRV $X$ is defined as the mathematical expectation of the square of the deviation of $X$ from its mathematical expectation:

$$
D(X)=M\left([X-M(X)]^{2}\right)
$$

The quantity $D(x)$ can be calculated using the formula:

$$
D(x)=M\left(x^{2}\right)-[M(x)]^{2}
$$

which follows from the definition of variance.

The main properties of the variance are expressed by the equalities:

1) $D(C)=0$
2) $D(C X)=C^{2} \cdot D(X)$
3) $D(X+Y)=D(X)+D(Y)$
4) if $X$ and $Y$ are independent, then $D(X Y)=D(X) \cdot D(Y)$.

The quantity $\sigma(X)=\sqrt{D(X)}$ is called the standard deviation and expresses the degree of spread of $X$ around $M(X)$. The variance of a random variable distributed according to a known law is expressed through the parameters of this law.

If the random variable $X$ is distributed according to the law:

- Bernoulli, then $D(X)=n p q$
- Poisson, then $D(X)=\lambda$
- geometric, then $D(X)=1 / p-p$
- hypergeometric, then $D(X)=\frac{N-n}{N-1} \cdot n \cdot \frac{M}{N}\left(1-\frac{M}{N}\right)$.

$5^{\circ}$ The initial moment of order $k$ of a DRV $X$ is defined as the mathematical expectation of the $k$-th power of this variable (4 moments are considered):

$$
M_{k}(X)=M\left(X^{k}\right)=\sum_{\imath=1}^{n} p_{\imath} x_{\imath}^{k}, \quad k=1,2,3,4
$$

In particular, $M_{1}(X)=M(X)$.

$6^{\circ}$. The DRV $X-M(X)$ is also called the centered DRV. The central moment of order $k$ of a DRV $X$ is defined as the initial moment of order $k$ of the DRV $X-M(X)$:

$$
m_{k}(X)=M[X-M(X)]^{k}=\sum p_{\imath}\left[x_{\imath}-M(X)\right]^{k}
$$

In particular, $m_{2}=D(X)$.

The following formulas hold:

$$
m_{1}=0, \quad m_{2}=M_{2}-M_{1}^{2}, \quad m_{3}=M_{3}-3 M_{1} M_{2}+2 M_{1}^{3}
$$

$7^{\circ}$. The skewness and kurtosis of a DRV $X$ are defined as the quantities

$$
A s(X)=\frac{m_{3}}{\sigma^{3}}, \quad E k(X)=\frac{m_{4}}{\sigma^{4}}-3
$$

$A s