Task 1. Given the interval distribution:

| Intervals | $(0 ; 5)$ | $(5,10)$ | $(10 ; 15)$ | $(15 ; 20)$ | $(20 ; 25)$ | $(25 ; 30)$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Midpoints | 2.5 | 7.5 | 12.5 | 17.5 | 22.5 | 27.5 |
| Frequencies | 133 | 45 | 15 | 4 | 2 | 1 |

It is required, at the significance level $\alpha=0.05$, using the Pearson's criterion, to confirm or refute the hypothesis that this distribution follows an exponential law.
Solution. 1) The empirical sample mean $\widetilde{x}_{B}$ is calculated using the centers of the intervals:

$\widetilde{x}_{\mathrm{B}}=\frac{1}{200}(133 \cdot 2.5+45 \cdot 7.5+15 \cdot 12.5+4 \cdot 17.5+2 \cdot 22.5+$ $+1 \cdot 27.5)=5$

We take $\lambda=\frac{1}{\widetilde{x}_{\mathrm{B}}}=\frac{1}{5}=0.2$. Thus, the density of the exponential distribution function is: $f(x)=0.2 e^{-0.2 x}$.

2) We calculate $p\left(x_{\imath}<X<x_{\imath+1}\right)=e^{-0.2 x_{1}}-e^{-0.2 x_{1+1}}=p_{\imath}$.

We get: $p_{1}=p(0<X<5)=e^{-0.2 \cdot 0}-e^{-0.2 \cdot 5}=1-e^{-1}=0.632$.

Similarly, $p_{2}=0.233 ; p_{3}=0.086 ; p_{4}=0.032 ; p_{5}=0.012 ; p_{6}=$ $=0.004$.

3) We calculate the theoretical frequencies $n_{k}^{\prime}=200 \cdot p_{i}$ :

$n_{1}^{\prime}=126.4 \approx 126 ; n_{2}^{\prime}=46.5 \approx 47 ; n_{3}^{\prime}=17.1 \approx 17 ; n_{4}^{\prime}=6.3 \approx$ $\approx 6 ; n_{5}^{\prime}=2.3 \approx 2 ; n_{6}^{\prime}=0.8 \approx 1$.

4) We apply the Pearson's criterion (frequencies $4+2+1$ are combined into one; the corresponding theoretical frequency is $6.3+2.3+0.8=$ $=9.4)$.

We get $\chi_{\text {obs }}^{2}=0.34+0.05+0.26+0.64=1.29$.

After combining three intervals, the number of degrees of freedom becomes $r=4-2=2$. From the table in the appendix, we take $\chi_{\text {cr }}^{2}=6.0$.

Answer. The hypothesis is confirmed.

## Exercise

It is required at the significance level $\alpha=0.01$ to confirm or reject the hypothesis that the given interval distribution follows an exponential law:

| Interval | $(0,10)$ | $(10,20)$ | $(20,30)$ | $(30,40)$ | $(40,50)$ | $(50,60)$ | $(60,70)$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Frequencies | 306 | 140 | 72 | 12 | 6 | 3 | 1 |

## § 8. Linear Correlation of Random Variables

$1^{\circ}$. The correlation dependence between random variables (features) $X$ and $Y$ is usually given using a correlation table:

|  |  |  |  |  |  |  | $Y$ |  |  |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  | $y_{1}$ | $y_{2}$ | .. | $y_{j}$ | $\ldots$ | $y_{m}$ | $n_{x}$ |  |  |  |  |  |  |
| $x_{1}$ | $n_{11}$ | $n_{12}$ | $\ldots$ | $n_{1 j}$ | $\ldots$ | $n_{1 m}$ | $n_{x_{1}}$ |  |  |  |  |  |  |
| $x_{2}$ | $n_{21}$ | $n_{22}$ | $\ldots$ | $n_{2 j}$ | $\ldots$ | $n_{2 m}$ | $n_{x_{2}}$ |  |  |  |  |  |  |
| .. | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ | .. | $\ldots$ | $\ldots$ |  |  |  |  |  |  |
| $x_{2}$ | $n_{\imath 1}$ | $n_{\imath 2}$ | $\ldots$ | $n_{\imath j}$ | $\ldots$ | $n_{\imath m}$ | $n_{x_{i}}$ |  |  |  |  |  |  |
| $\ldots$ | $\ldots$ | $\ldots$ | .. | $\ldots$ | $\ldots$ | $\ldots$ | $\ldots$ |  |  |  |  |  |  |
| $x_{k}$ | $n_{k 1}$ | $n_{k 2}$ | .. | $n_{k_{3}}$ | $\ldots$ | $n_{k m}$ | $n_{x_{k}}$ |  |  |  |  |  |  |
| $n_{y}$ | $n_{y_{1}}$ | $n_{y_{2}}$ | . | $n_{y_{1}}$ | $\ldots$ | $n_{y_{n n}}$ | $n$ |  |  |  |  |  |  |

Here $x_{1}, x_{2}, \ldots, x_{\imath}, \ldots, x_{k} ; y_{1}, y_{2}, \ldots, y_{j}, \ldots, y_{m}-$ are the midpoints of the intervals or features of $X$ and $Y$, and $n_{x_{1}}, n_{x_{2}}, \ldots, n_{x_{2}}, \ldots, n_{x_{k}} ; n_{y_{1}}, n_{y_{2}}$, ..., $n_{y_{2}}, \ldots, n_{y_{m}}$ are the corresponding frequencies, $n_{\imath \jmath}$ is the frequency of the pair $\left(x_{i}, y_{i}\right) ; n=\sum_{i=1}^{k} \sum_{j=1}^{m} n_{\imath j}$.

$2^{\circ}$. The correlation dependence between features $X$ and $Y$ given by the table can be replaced by a functional dependence if each value $x_{\imath}$ of feature $X$ is associated with the mean value of feature $Y$, i.e., the quantity

$$
\bar{Y}_{x_{\imath}}=\frac{\sum_{j=1}^{n} n_{\imath \jmath} y_{j}}{n_{x_{،}}}
$$

- the mean value of $Y$.

If now the points ( $x_{\imath}, \bar{Y}_{x_{\imath}}$ ) are aligned along the curve $y=f\left(x, \alpha_{1}, \alpha_{2}, \ldots, \alpha_{m}\right)$ using the method of least squares, then the latter is called the regression line of $Y$ on $X$, and its equation is the regression equation of $Y$ on $X$. Similarly, the regression line of $X$ on $Y$ is defined.

The simplest and most important cases of regression lines are straight lines.

$3^{\circ}$. If both the regression line of $Y$ on $X$ and the regression line of $X$ on $Y$ are straight lines, then the quantities $Y$ and $X$ are called linearly correlated.

4. The sample equation of the straight regression line of $Y$ on $X$ is:

$$
\bar{y}_{x}-\bar{y}_{\mathrm{B}}=r_{\mathrm{B}} \frac{\sigma_{y}}{\sigma_{x}}\left(x-\bar{x}_{\mathrm{B}}\right)
$$

where $\bar{y}_{x}$ is the group mean,

$\bar{y}_{\mathrm{B}}, \bar{x}_{\mathrm{B}}$ are the sample means of features $Y$ and