2020 EJC P2 Q9

A computer chip generates a random number from the set \(\left\{ 1,2,3,4 \right\}\). The probability distribution of the number generated, \(X\), is given by

\(\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix} p\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=1 \\ \frac{1-p}{3}\,\,\,\,\,\,\,\,\,\,x=2,3,4 \\ \end{matrix} \right.\)

where \(0 < p < 1\).

1. Find \(\mathrm{E}\left( X \right)\) in terms of \(p\) and hence show that \(\mathrm{Var}\left( X \right)=\frac{2}{3}\left( 1-p \right)\left( 1+6p \right)\).[3]
2. \({{X}_{1}}\) and \({{X}_{2}}\) are two independent observations of \(X\). Find \(\mathrm{P}\left( \left| {{X}_{1}}-{{X}_{2}} \right|>1 \right)\) in terms of \(p\). Hence, find the maximum possible value of this probability as \(p\) varies.[5]

It is given instead that \(p=\frac{1}{2}\).

3. Find the probability that the sum of \(50\) independent observations of \(X\) exceeds \(110\).[3]