15.6.15 Definition and properties of probability
Problem:
The distribution density of the random variable \( X \) has the form
\[
f(x)=\left\{\begin{array}{rc}
0, & \text { when } x<1 ; \\
\frac{a}{x^{3}}, & \text { when } 1 \leq x \leq 4 ; \\
0, & \text { when } x>4
\end{array}\right.
\]
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