Math 6010 • Linear Models

A random experiment has exactly three possible outcomes, referred
to as outcomes 1, 2, and 3, with probabilities \(p_1>0\), \(p_2>0\), and \(p_3>0\),
where \(p_1+p_2+p_3=1\). We consider a sequence of independent trials, at each of
which the specified random experiment is performed. For \(i=1,2\), let \(N_i\) be
the number of trials needed for outcome \(i\) to occur, and put
\(N:=\min(N_1,N_2)\).

\textbf{(a)} Show that \(N\) is independent of \(1_{\{N_1<N_2\}}\).

\textbf{(b)} Evaluate \(\E[N_1\mid N_1<N_2]\).

\textbf{(c)} Roll a pair of dice until a total of 6 or 7 appears. Given that 6
appears before 7, what is the (conditional) expected number of rolls?