IB AA HL May 2026 P2 TZC Q7

The continuous random variable \(X\) has probability density function

\[f(x)=\begin{cases}\frac34x^2(2-x),&0\leq x\leq2,\\0,&\text{otherwise}.\end{cases}\]

1. Find \(P(0.5\leq X<1.5)\).[2]
2. The discrete random variable \(Y\) is the value of \(X\) rounded to the nearest integer. The probability distribution of \(Y\) is shown in the following table.[3]

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

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