2 The continuous random variable \(X\) takes values in the interval \(- 1 \leq x \leq 1\) and has probability density function
$$f ( x ) = \left\{ \begin{array} { l r }
a & - 1 \leq x < 0
a + x ^ { 2 } & 0 \leq x \leq 1
\end{array} \right.$$
where \(a\) is a constant.
- (A) Sketch the probability density function.
(B) Show that \(a = \frac { 1 } { 3 }\). - Find
(A) \(\mathrm { P } \left( X < \frac { 1 } { 2 } \right)\),
(B) the mean of \(X\). - Show that the median of \(X\) satisfies the equation \(2 m ^ { 3 } + 2 m - 1 = 0\).