2018 JJC P2 Q6

Alice and Betty each throw a fair cubical die simultaneously.

The random variable \(X\) is the larger number shown on the two dice or the common number of the dice if the numbers are equal.

1. Show that \(\mathrm{P}\left( X\le x \right)={{\left( \frac{x}{6} \right)}^{2}}\) for \(x=1,2,...,6.\)[2]
2. Deduce that \(\mathrm{P}\left( X=x \right)=\frac{2x-1}{36}\) for \(x=1,2,...,6.\)[1]
3. Show that \(\mathrm{E}\left( X \right)=\frac{161}{36}\) and \(\mathrm{Var}\left( X \right)=\frac{2555}{1296}\).[3]
4. Forty independent observations of \(X\) are taken. Using a suitable approximation, estimate the probability that the mean of these observations is at least \(4.5\).[3]