15.6.15 Definition and properties of probability

Problem: The distribution density of the random variable $X$ has the form $$f(x)=\{\begin{array}{rc}0,& \text{ when }x<1;\\ \frac{a}{x^3},& \text{ when }1\le x\le 4;\\ 0,& \text{ when }x>4\end{array}$$ Find: a) the coefficient $a$; b) the distribution function $F(x)$; c) the expected value $E(X)$ and the dispersion $D(X)$; d) the probability \( P(3