Example. A die is rolled. Let $\xi$ be the number of points that come up. Find the mathematical expectation and variance of the random variable $\eta=\min (\xi, 3)$.

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Translation:
Example. A die is rolled. Let $\xi$ be the number of points that come up. Find the mathematical expectation and variance of the random variable $\eta=\min (\xi, 3)$.
## Solution.

1. We find the distribution series of the random variable $\eta$.

The elementary outcome of the considered random experiment is the number of points fallen $i$: $i=1, \ldots, 6$. The random variable $\eta$ on the $i$-th elementary outcome takes the value $\min (i, 3)$. We construct a table of elementary outcomes and values of $\eta$

| $i$ | 1 | 2 | 3 | 4 | 5 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\eta$ | 1 | 2 | 3 | 3 | 3 | 3 |

Since all elementary outcomes are equally likely, the distribution series of the random variable $\eta$ is as follows

| $\eta$ | 1 | 2 | 3 |
| :---: | :---: | :---: | :---: |
| $\mathrm{P}$ | $1 / 6$ | $1 / 6$ | $4 / 6$ |

2. We find the mathematical expectation $M \eta$ of the discrete random variable $\eta$ using formula (1). We get

$$
\mathrm{M} \eta=1 \cdot \frac{1}{6}+2 \cdot \frac{1}{6}+3 \cdot \frac{4}{6}=\frac{15}{6}=\frac{5}{2}
$$

3. We find the variance $\mathrm{D} \eta$ of the discrete random variable $\eta$ using formula (3). We get

$$
\mathrm{D} \eta=\left(1-\frac{5}{2}\right)^{2} \frac{1}{6}+\left(2-\frac{5}{2}\right)^{2} \frac{1}{6}+\left(3-\frac{5}{2}\right)^{2} \frac{4}{6}=\frac{7}{12}
$$

Answer. $\mathrm{M} \eta=5 / 2, \mathrm{D} \eta=7 / 12$.

### Conditions of the Problem. A die is rolled. Find the mathematical expectation and variance of the random variable $\eta(\xi)$, where $\xi-$ is the number of points fallen.

1. $\eta=\xi^{2}-11 \xi+30$.
2. $\eta=(\xi-3)^{2}$.
3. $\eta=\xi^{2}-7 \xi+12$.
4. $\eta=|\xi-3|$.
5. $\eta=\max (\xi, 4)$.
6. $\eta=(\xi-2)(\xi-3)(\xi-4)$.
7. $\eta=(\xi-1)(\xi-2)(\xi-5)(\xi-6)$.
8. $\eta=\sin (\pi \xi / 2)$.
9. $\eta=\cos (\pi \xi / 2)$.
10. $\eta=\sin [\pi(\xi-3) / 6]$.
Answers.
11. $\mathrm{M} \eta=20 / 3, \mathrm{D} \eta=949 / 18 . \quad$ 2. $\quad \mathrm{M} \eta=19 / 6, \quad \mathrm{D} \eta=329 / 36$.
12. $\mathrm{M} \eta=8 / 3, \quad \mathrm{D} \eta=56 / 9 . \quad$ 4. $\quad \mathrm{M} \eta=3 / 2, \quad \mathrm{D} \eta=11 / 12$.
13. $\mathrm{M} \eta=9 / 2, \quad \mathrm{D} \eta=7 / 12$.
14. $\mathrm{M} \eta=4$,

$\mathrm{D} \eta=92$.
7. $\mathrm{M} \eta=4$

$\mathrm{D} \eta=32$.
8. $\mathrm{M} \eta=1 / 6$,

$\mathrm{D} \eta=17 / 36$
9. $\mathrm{M} \eta=-1 / 6, \mathrm{D} \eta=17 / 36$.
10. $\mathrm{M} \eta=1 / 6, \quad \mathrm{D} \eta=17 / 36$.

### 6.15. Numerical Characteristics of a Continuous Random Variable

### Problem Statement. Find the mathematical expectation M $\eta$, variance $\mathrm{D} \eta$, and probability $\mathrm{P}(\eta>c)$, where $c$ is a constant, of a continuous random variable $\eta$ with a given probability density function $p(x)$.

### Solution Plan. The mathematical expectation M $\eta$ of a continuous random variable $\eta$ is defined by the formula

$$
\mathrm{M} \eta=\int_{-\infty}^{+\infty} x p(x) d x
$$

(assuming the integral converges absolutely).

The variance $\mathrm{D} \eta$ of the random variable $\eta$ is defined by the formula

$$
\mathrm{D} \eta=\mathrm{M}(\eta-\mathrm{M} \eta)^{2}
$$

For finding the variance when $\eta$ is a continuous random variable, the formula used is

$$
\mathrm{D} \eta=\int_{-\infty}^{+\infty}(x-m)^{2} p(x) d x=\int_{-\infty}^{+\infty} x^{2} p(x) d x-m^{2}
$$

where $m=\mathrm{M} \eta$.

From the definition of the probability density function, it follows that

$$
\mathrm{P}(\eta>c)=\int_{c}^{+\infty} p(x) d x
$$

Remark. If the density $p(x)$ is given up to some parameter, its value can be found from the relation

$$
\int_{-\infty}^{+\infty} p(x) d x=1
$$

1. We find the analytical expression for the probability density function $p(x)$ of the random variable $\eta$.
2. We find the mathematical expectation $M \eta$ of the continuous random variable $\eta$ using formula (1).
3. We find the variance $\mathrm{D} \eta$ of the continuous random variable $\eta$ using formula (3).
4. We find $\mathrm{P}(\xi>c)$ using formula (4).