Question: The probability density of the random variable Y is defined as follows:

\(f\left( y \right)=\left\{ \begin{aligned} & ky\left( 3+2y-{{y}^{2}} \right)\,\,\,\,\,\,\,\,\,\,\,0\le y\le 3 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ otherwise} \\ \end{aligned} \right.\)

(a) Find the value k. Then graph f(y) against y, clearly and neatly.

(b) Find E(Y)

(c) Compute the variance of Y.

(d) Find the cumulative distribution function F(y). Graph F(y) against y, clearly and neatly. Use this graph to obtain an approximate value of the median of this distribution.