N1998 P2 Q7

A computer can give independent observations of a random variable \(X\) with probability distribution given by \(\mathrm{P}\left( X=0 \right)=\frac{3}{4}\) and \(\mathrm{P}\left( X=2 \right)=\frac{1}{4}\). It is programmed to output a value for the random variable \(Y\) defined by \(Y={{X}_{1}}+{{X}_{2}}\), where \({{X}_{1}}\) and \({{X}_{2}}\) are independent observations of \(X\). Tabulate the probability distribution of \(Y\), show that \(E\left( Y \right)=1\).[3]

The random variable \(T\) is defined by \(T={{Y}^{2}}\). Find \(\mathrm{E}\left( T \right)\) and show that \(\mathrm{Var}\left( T \right)=\frac{63}{4}\).[3]

The computer is programmed to produce a large number \(n\) of independent values of \(T\) and to calculate the mean \(M\) of these values. Find the smallest value of \(n\) such that \(\mathrm{P}\left( M<3 \right)>0.99\).[4]