6 Coursework Exercises 2

• C2.1

  Consider a duel. Players $A$ and $B$ take it in turns to shoot at each other. Player $A$ shoots first, and has probability $a$ of hitting player $B$. At his turn, player $B$ has probability $b$ of shooting player $A$. The winner is the first player to hit the other.

  Let $N$ be the number of shots in the duel. Show that the pgf of $N$ is given by

  $$G(z)=\frac{az+b(1-a)z^2}{1-z^2(1-a)(1-b)}.$$

  Hence calculate the mean of $N$.

  [3]

• C2.2

  The pgf of the time to the first success in a Bernoulli process with success probability $p$ is

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

  Let $X$ be the time until the first success in a sequence of Bernoulli trials with success probability $1/3$, and $Y$ the time until the first success in a sequence of Bernoulli trials with success probability $2/3$. Write down the pgf for $T=X+Y$, assuming the two sequences of Bernoulli trials are independent.

  Using the same approach as in 3.2.2 in the notes, show that if $p_i=\text{P }(T=i)$ then for $i>2$:

  $$9p_i-9p_{i-1}+2p_{i-2}=0$$

  Solve this difference equation (using the correct initial conditions) to obtain $p_i$.

  [5]

• C2.3

  Consider the pgf of $N$ in question 1 of this sheet. Denote this pgf by $G(z)$.

  Calculate $\frac{1}{2}(G(1)-G(-1))$. How does this relate to the probability that player A wins the duel (see quiz sheet 1).

  Let $p_i=\text{P }(N=i)$. By expanding the probability generating function:

  $$G(1)=p_0+p_1+p_2+\mathrm{\dots },\text{ and }G(-1)=p_0-p_1+p_2-\mathrm{\dots },$$

  explain why the relationship between $\frac{1}{2}(G(1)-G(-1))$ and the probability that player A wins holds.

  [2]

Full marks may be obtained for correct solutions to the previous questions. A further 5 marks, up to maximum total mark of 10, can be gained by successfully answering the following.

• C2.4

  For the rv $T$, the time to ruin starting from $X_0=1$ in a random walk with $p=\frac{1}{2}$, show using the pgf of $T$ that

  $$P(T=2m+1)=\frac{(2m)!}{m!(m+1)!}{\left(\frac{1}{2}\right)}^{(2m+1)}.$$