2026 RI P2 Q5

A random variable \(X\) has probability distribution given by

\[\mathrm{P}(X=x)=\begin{cases}kx^2&\text{for}\hspace{0.5em}x=1,2,3,\\k(7-x)&\text{for}\hspace{0.5em}x=4,5,6,\\0&\text{otherwise},\end{cases}\]

where \(k\) is a positive constant.

1. Show that \(k=\dfrac{1}{20}\).[2]
2. Find the values of \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\).[3]

Another random variable \(Y\) is defined by \(Y=3X_1-X_2\), where \(X_1\) and \(X_2\) are independent observations of \(X\).

3. Find \(\mathrm{Var}(Y)\).[2]