87. A discrete random variable $X$ is given by the distribution function

$$
F(x)=\left\{\begin{array}{lll}
0 & \text { if } & x \leq 2 \\
0.5 & \text { if } & 2 < x \leq 6 \\
1 & \text { if } & x > 8
\end{array}\right.
$$

Find the probability that $X$ will take a value not less than 4 and less than 8.
Solution. The required probability $P(4 \leq x<8)$ will be found using formula (28):

$$
P(4 \leq x<8)=F(8)-F(4)=0.7-0.5=0.2 .
$$

## 4.2. Numerical Characteristics of Discrete Random Variables

The mathematical expectation of a discrete random variable $X$, having a finite number of possible values, is equal to

$$
M(X)=\sum_{i=1}^{n} x_{i} p_{i}
$$

The mathematical expectation of a discrete random variable $X$, having a countable set of values, is equal to

$$
M(X)=\sum_{i=1}^{\infty} x_{i} p_{i}
$$

provided that the series on the right-hand side converges absolutely.

In formulas (29) and (30): $x_{i}$ - possible values of the random variable $X$, $p_{i}$ - probabilities that the random variable $X$ will take these values.

Properties of mathematical expectation:

$1^{\circ} . M(C)=C$, where $C$ is a constant.

$2^{\circ} . M(k X)=k M(X)$

where $k=$ const.

$3^{\circ} \cdot M(X+Y)=M(X)+M(Y)$

where $X$ and $Y$ are any two random variables.

$4^{\circ} . M(X Y)=M(X) \cdot M(Y)$,

where $X$ and $Y$ are two independent random variables.

The variance of a random variable $\mathrm{X}$ is defined by the equality

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

or the equivalent equality

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

The variance of a discrete random variable, having a finite number of possible values, can be calculated using the formula

$$
D(X)=\sum_{i=1}^{n}\left(x_{i}-M(X)\right)^{2} \cdot p_{i}
$$

corresponding to formula (35), or using the formula

$$
D(X)=\sum_{i=1}^{n} x_{i}^{2} p_{i}-M^{2}(X)
$$

corresponding to formula (36).

Properties of variance:

$$
\begin{aligned}
& 1^{\circ} . D(C)=0 \\
& 2^{\circ} . D(k X)=k^{2} D(X) \\
& 3^{\circ} . D(X+Y)=D(X)+D(Y)
\end{aligned}
$$

where $X$ and $Y$ are two independent random variables.

The mean square deviation (standard deviation) of a random variable $\boldsymbol{X}$ is equal to

$$
\sigma(X)=\sqrt{D(X)}
$$