15.2.14 One dimensional random variables and their characteristics

Problem: Random variables \( \tau \) and \( \xi \) are functionally related to each other: \( \tau=1 / \xi \). It's known that the random variable \( \xi \) is continuous and has the following distribution density: \[ f_{\xi}(x)=\left\{\begin{array}{ll} 0, & \text { if } x<1 \\ 3 / x^{4}, & \text { if } x \geq 1 \end{array}\right. \] Is it possible to claim that the random variable \( \tau \) will also be continuous? Justify the answer. Find the expression of the density \( f_{\tau}(z) \). Calculate the expected value, the dispersion \( E[\tau], V[\tau] \) and the possibility \( p\{0,1<\tau<0,3\} \).