2 Quiz Exercises 1

Let $T$ be the time at which the first success $S$ is observed in a sequence of Bernoulli trials. Each Bernoulli trial has a success probability of $p$.

For Questions 1–3 the possible answers are:

• (a)

  $1/p$;

• (b)

  $1/q$;

• (c)

  $q/p^2$;

• (d)

  $p/q^2$; or

• (e)

  None of the above.

• Q1.1

  Bernoulli 1: What is $\text{E}(T)$?

• Q1.2

  Bernoulli 2: What is $\text{E}(T^2)$?

• Q1.3

  Bernoulli 3: What is $\text{Var}(T)$?

Hint: Adapt Exercise 1.3.2 from the notes. For Q1.2 use $T^2$ in place of $T$. (You will need to expand the expression $E[{(1+T^{\prime })}^2]$ where $T^{\prime }$ has the same distribution as $T$.)

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.

By conditioning on the first two shots solve the following:

• Q1.4

  Duel 1: Is $\text{P}(A\text{ wins})$:

  • (a)

    $a/(1-(1-a)(1-b))$;

  • (b)

    $b/(1-(1-a)(1-b))$;

  • (c)

    $a$;

  • (d)

    $b$; or

  • (e)

    None of the above?

• Q1.5

  Duel 2: Is the expected number of shots in the duel:

  • (a)

    $1/(1-(1-a)(1-b))$;

  • (b)

    $a/(1-(1-a)(1-b))$;

  • (c)

    $(2-a)/(1-(1-a)(1-b))$;

  • (d)

    $(1+a)/(1-(1-a)(1-b))$; or

  • (e)

    None of the above?