15.2.48 One dimensional random variables and their characteristics

Problem: The distribution density of the random variable \( X \) is given. \[ f(x)=\left\{\begin{array}{cl} 0, & \text { when } x<0 \\ \alpha\left(x-\frac{x^{2}}{3}\right), & \text { when } 0 \leq x \leq 3 \\ 0, & \text { when } x>3 \end{array}\right. \] Find a) the coefficient \( \alpha \), b) the distribution function \( F(x) \), c) the expected value \( E(X) \) and the dispersion \( D(X) \).