88. A discrete random variable $X$ is given by the distribution series:

$$
X: \begin{array}{cccc}
x_{i} & 1 & 2 & 4 \\
p_{i} & 0.1 & 0.3 & 0.6^{\circ}
\end{array}
$$

Find its mathematical expectation, variance, and standard deviation.
Solution. The expected value of the random variable $X$ will be found using formula (29):

$$
M(X)=x_{1} p_{1}+x_{2} p_{2}+x_{3} p_{3}=1 \cdot 0.1+2 \cdot 0.3+4 \cdot 0.6=3.1
$$

The variance of $X$ will be calculated using formula (37) and formula (38). Applying formula (37), we get

$$
\begin{aligned}
& D(X)=\sum_{i=1}^{n}\left[x_{i}-M(X)\right]^{2} p_{i}= \\
& =(1-3.1)^{2} \cdot 0.1+(2-3.1)^{2} \cdot 0.3+(4-3.1)^{2} \cdot 0.6=1.29
\end{aligned}
$$

Applying formula (38), we get

$$
\begin{aligned}
& D(X)=\sum_{i=1}^{n} x_{i}^{2} p_{i}-M^{2}(X)= \\
& =1^{2} \cdot 0.1+2^{2} \cdot 0.3+4^{2} \cdot 0.6-3.1^{2}=1.29
\end{aligned}
$$

Comparison shows that in this problem, the variance is easier to calculate using formula (38) than using formula (37).

The standard deviation of the random variable $X$ will be found using formula (43):

$$
\sigma(X)=\sqrt{D(X)}=\sqrt{1.29} \approx 1.1358
$$