2022 TMJC P2 Q6

The probability function of a discrete random variable, \(X\), is given as follows:

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

1. Given that the expected value of \(X\) is \(2\), find the probability distribution of \(X\).[4]
2. Given that \({{X}_{1}}\) and \({{X}_{2}}\) are two independent observations of \(X\), find \(\mathrm{P}\left( {{X}_{1}}+{{X}_{2}}=4 \right)\).[2]