A random variable \(\xi\) is distributed according to the logarithmic normal law; i.e., for \(x>0\) the density function of \(\xi\) is

\[ p(x)=\frac{1}{x \beta \sqrt{2 \pi}} e^{-\frac{1}{2 \beta^{2}}(\ln x-\alpha)^{2}} \]

\((p(x)=0\) for \(x \leqslant 0)\). Find \(M \xi\) and \(D \xi\).

(A. N. Kolmogorov has demonstrated that particle sizes in crushing obey the logarithmic normal distribution law.)