91Ó°ÊÓ

Let \(X\) and \(Y\) be independent geometric random variables with respective parameters \(\alpha\) and \(\beta\). Show that $$ \mathrm{P}(X+Y=z)=\frac{\alpha \beta}{\alpha-\beta}\left\\{(1-\beta)^{z-1}-(1-\alpha)^{z-1}\right\\} $$

Show that the sum of two independent binomial variables, \(\operatorname{bin}(m, p)\) and bin \((n, p)\) respectively, is bin \((m+n, p)\)

You roll a conventional fair die repeatedly. If it shows 1 , you must stop, but you may choose to stop at any prior time. Your score is the number shown by the die on the final roll. What stopping strategy yields the greatest expected score? What strategy would you use if your score were the square of the final roll?

\(N+1\) plates are laid out around a circular dining table, and a hot cake is passed between them in the manner of a symmetric random walk: each time it arrives on a plate, it is tossed to one of the two neighbouring plates, each possibility having probability \(\frac{1}{2}\). The game stops at the moment when the cake has visited every plate at least once. Show that, with the exception of the plate where the eake began, each plate has probability \(1 / N\) of being the last plate visited by the cake.