Consider the Following Function $$f ( x ) = \left\{ \begin{array} { l l } 0 & x \leq 0 \\\frac { x ^ { 3 } } { 4 } + c & 0 \leq x \leq 2 \\0 & x > 2\end{array} \right.$$

Short Answer

Consider the following function: $$f ( x ) = \left\{ \begin{array} { l l } 0 & x \leq 0 \\\frac { x ^ { 3 } } { 4 } + c & 0 \leq x \leq 2 \\0 & x > 2\end{array} \right.$$
(a) Find the value of so that is a probability density function.
(b) Let be a random variable with this probability density function. Find the probability that is between 1 and 1.5.
(c) Find the probability that the function $g ( X ) = X ^ { 2 } - 2 X$ is increasing.