4 Workshop Exercises 2

A tennis game has reached deuce. The server has probability $p$ of winning each subsequent point (otherwise the receiver wins the point). Each point is independent of all others, and to win the game either the server or receiver needs to be two points clear.

Let $T$ be the number of subsequent points played until the game finishes. Show that the pgf of $T$ is

$$G_T(z)=\frac{(p^2+q^2)z^2}{1-2pq{z}^2}.$$

(1)

In the case of $p=1/2$, calculate $\text{E}(T)$.

Consider the pgf from the previous question in the case $p=1/2$. Let $p_i$ be the probability that $T=i$. By re-writing (1) as

$$\left(1-\frac{1}{2}{z}^2\right)\left(p_0+p_{1}z+\mathrm{\cdots }+p_i{z}^i+\mathrm{\cdots }\right)=\frac{1}{2}{z}^2,$$

and by equating coefficients of $z$, $z^2$ and, for $i\ge 1$, $z^{i+2}$, calculate $p_1$, $p_2$ and show that

$$p_{i+2}-\frac{1}{2}{p}_i=0,\text{ for }i=1,2,\mathrm{\dots }.$$

Solve this difference equation to find the solution for $p_i$.

By conditioning on the outcome of the first trial, show that the pgf of the time $T$ at which the first success occurs in a sequence of Bernoulli trials is

$$G(z)=\frac{pz}{1-qz}.$$

From this derive the mean and variance of $T$.

A coin with probability $p$ of showing heads is tossed repeatedly. Find the probability generating function for the number $T$ of tosses before a run of $n$ consecutive heads has appeared for the first time.

A fisherman catches $N$ fish where $N$ is Poisson with parameter $\lambda$. The weight of the $n$-th fish is $W_n$ where $W_1,W_2,\mathrm{\dots }$ are independent identically distributed random variables with common probability generating function $G(z)$. By conditioning on the value of $N$, show that the generating function $G_W(z)$ of the total weight of fish caught $W={\sum }_{i=1}^N{W}_i$ is given by

$$G_W(z)=\exp (-\lambda +\lambda G(z)).$$