2021 TJC P2 Q9

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

\(\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix} kx\left( 6-x \right),\,\,\,x=1,2,3,4,5 \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{otherwise}\mathrm{.} \\ \end{matrix} \right.\)

1. Show that \(k=\frac{1}{35}\).[1]
2. Find the mean and variance of \(X\).[3]

\(40\) independent observations of \(X\) are taken.

3. Find the probability that the sum of the \(40\) observations of \(X\) exceeds \(110\).[3]
4. Find the probability that the maximum value observed is \(5\).[3]