Problems

\(A\) and \(B\) both toss the same biased coin. The probability that the coin shows heads is \(p\), where \(0 < p < 1\), and the probability that it shows tails is \(q = 1 - p\). Let \(X\) be the number of times \(A\) tosses the coin until it shows heads. Let \(Y\) be the number of times \(B\) tosses the coin until it shows heads.

1. The random variable \(S\) is defined by \(S = X + Y\) and the random variable \(T\) is the maximum of \(X\) and \(Y\). Find an expression for \(\mathrm{P}(S = s)\) and show that \[ \mathrm{P}(T = t) = pq^{t-1}(2 - q^{t-1} - q^t). \]
2. The random variable \(U\) is defined by \(U = |X - Y|\), and the random variable \(W\) is the minimum of \(X\) and \(Y\). Find expressions for \(\mathrm{P}(U = u)\) and \(\mathrm{P}(W = w)\).
3. Show that \(\mathrm{P}(S = 2 \text{ and } T = 3) \neq \mathrm{P}(S = 2) \times \mathrm{P}(T = 3)\).
4. Show that \(U\) and \(W\) are independent, and show that no other pair of the four variables \(S\), \(T\), \(U\) and \(W\) are independent.

Integers \(n_{1},n_{2},\ldots,n_{r}\) (possibly the same) are chosen independently at random from the integers \(1,2,3,\ldots,m\). Show that the probability that \(\left|n_{1}-n_{2}\right|=k\), where \(1\leqslant k\leqslant m-1\), is \(2(m-k)/m^{2}\) and show that the expectation of \(\left|n_{1}-n_{2}\right|\) is \((m^{2}-1)/(3m)\). Verify, for the case \(m=2\), the result that the expection of \(\left|n_{1}-n_{2}\right|+\left|n_{2}-n_{3}\right|\) is \(2(m^{2}-1)/(3m).\) Write down the expectation, for general \(m\), of \[ \left|n_{1}-n_{2}\right|+\left|n_{2}-n_{3}\right|+\cdots+\left|n_{r-1}-n_{r}\right|. \] Desks in an examination hall are placed a distance \(d\) apart in straight lines. Each invigilator looks after one line of \(m\) desks. When called by a candidate, the invigilator walks to that candidate's desk, and stays there until called again. He or she is equally likely to be called by any of the \(m\) candidates in the line but candidates never call simultaneously or while the invigilator is attending to another call. At the beginning of the examination the invigilator stands by the first desk. Show that the expected distance walked by the invigilator in dealing with \(N+1\) calls is \[ \frac{d(m-1)}{6m}[2N(m+1)+3m]. \]