5. Why are the parameters $a$ and $b$ of a uniformly distributed random variable $X$ estimated by the formulas

$$
a^{*}=\bar{x}_{\mathrm{B}}-\sqrt{3} \sigma_{\mathrm{B}}, \quad b^{*}=\bar{x}_{\mathrm{B}}+\sqrt{3} \sigma_{\mathrm{B}} ?
$$
Solution. It is known that the sample mean $\bar{x}_{\mathrm{B}}$ and the sample standard deviation $\sigma_{\mathbf{z}}$ can be used as estimates for the expected value and standard deviation of a random variable $X$, respectively.

It is also known (see Chapter VI, problems 313, 315) that for a uniform distribution, the expected value and standard deviation are respectively:

$M(X)=(a+b) / 2, \quad \sigma(X)=\sqrt{D(X)}=\sqrt{(b-a)^{2}}, \quad 12=(b-a) / 2 \sqrt{3}$.

Therefore, for estimating the parameters of a uniform distribution, we obtain the system of two linear equations:

$$
\left\{\begin{array} { l } 
{ ( b ^ { * } - a ^ { * } ) / 2 = \overline { x } _ { \mathrm { B } } , } \\
{ ( b ^ { * } - a ^ { * } ) / 2 \sqrt { 3 } = \sigma _ { \mathrm { B } } , }
\end{array} \quad \text { or } \quad \left\{\begin{array}{l}
b^{*}+a^{*}=2 \overline{x_{\mathrm{B}}}, \\
b^{*}-a^{*}=2 \sqrt{3} \sigma_{\mathrm{B}} .
\end{array}\right.\right.
$$

Solving this system, we get $a^{*}=\bar{x}_{\mathrm{B}}-\sqrt{3} \sigma_{\mathrm{B}}, b^{*}=\bar{x}_{\mathrm{B}}+\sqrt{3} \sigma_{\mathrm{B}}$.