2017 IJC BT2 Q7

The discrete random variable \(X\) has probability function

\(\mathrm{P}\left( X=x \right)=\left\{ \begin{matrix} k\left( \frac{5}{2}-x \right),\,\,\,x=0,2, \\ k\left( x-2 \right),\,\,\,\,\,\,\,\,\,\,x=3, \\ 0,\mathrm{otherwise,} \\ \end{matrix} \right.\)

where \(k\) is a positive constant.

1. Show that \(k=\frac{1}{4}\).[1]
2. Find \(\mathrm{E}\left( X \right)\) and show that \(\mathrm{Var}\left( X \right)=\frac{7}{4}\).[3]
3. A discrete random variable \(Y\) has \(\mathrm{E}\left( Y \right)=2\) and \(\mathrm{Var}\left( Y \right)=\frac{24}{5}\). Find the values of \(\mathrm{E}\left( 2X-3Y \right)\) and \(\mathrm{Var}\left( 2X-3Y \right)\), stating any assumption necessary for the calculations to be valid.[5]

Given that \({{X}_{1}}\) and \({{X}_{2}}\) are two independent observations of \(X\),

4. find the probability distribution of \(\left| {{X}_{1}}-{{X}_{2}} \right|\),[3]
5. find the probability that the sum of \({{X}_{1}}\) and \({{X}_{2}}\) exceeds \(4\), given that the sum is at least \(3\).[3]