Probabilities from a quadratic density function and the distribution and expectation of its rounded value

The probability density function of a continuous random variable \(X\) is \(f\), where

\[f(x)=\begin{cases}\frac{3}{16}x(4-x), & 0\leq x\leq2,\\ 0, & \text{otherwise}.\end{cases}\]

1. Find \(P(0.5\leq X<1.5)\).[2]
2. The discrete random variable \(Y\) is obtained by rounding \(X\) to the nearest whole number. Its probability distribution is given in the table below.[3]

   \(y\)	\(0\)	\(1\)	\(2\)
   \(P(Y=y)\)	\(a\)	\(b\)	\(c\)

   1. Write down \(b\).
   2. Find \(a\) and \(c\).
3. Find \(E(Y)\).[2]