15.2.27 One dimensional random variables and their characteristics

Problem: Random variables \( \tau \) and \( \xi \) are connected to each other functionally: \( \tau=\arctan \xi \). It's known that the random variable \( \xi \) is continuous, with the next distribution density: \[ f_{\xi}(x)=\left\{\begin{array}{cc} 0, & \text { if } x<2 \\ c / x^{5}, & \text { if } x \geq 2 \end{array}\right. \] 1. Find the constant \( c \). 2. If we can claim that the random variable \( \tau \) will be continuous, find the expression for the density of \( f_{\tau}(z) \). 3. Write the expression for the distribution function of the random variable \( \tau \). 4. Calculate \( E[\tau], V[\tau] \) and the probability of \( p\{2<\tau<10\} \).