problem 62

(a)

I_{k} = {\begin{matrix} 1 & if k sets a low or high record 0 & if k doesn’t set a low or high record \end{matrix}

(5.54)

P (I_{1}) = 1

P (I_{2}) = 1

P (I_{3}) = \frac{2}{3}

P (I_{4}) = \frac{2}{4}

and so on.

Now $I = I_{1} + \dots + I_{n}$
$E [I] = 1 + 1 + \frac{2}{3} \dots \frac{2}{100}$

(b)

I_{k} = {\begin{matrix} 1 & if k sets a low followed by a high 0 & otherwise \end{matrix}

(5.55)

P (I_{k}) = \frac{1}{k} \frac{1}{k + 1}

Now $I = I_{1} + \dots + I_{n}$
$E [I] = \frac{1}{1 \cdot 2} + \dots + \frac{1}{100 \cdot 101}$
$E [I] = 1 - \frac{1}{101}$

(c)

\begin{matrix} P (N > n) & = P (all of 2 to n + 1 fall short of 1) & (5.56) = \frac{1}{n + 1} & (5.57) \end{matrix}

\begin{matrix} P (N = n) & = P (N > n - 1) - P (N > n) & (5.58) = \frac{1}{n (n + 1)} & (5.59) \end{matrix}

(d)

\begin{matrix} E [N] & = \sum_{i = 1}^{\infty} iP (N = i) & (5.60) = \sum_{1}^{\infty} \frac{1}{i + 1} & (5.61) \end{matrix}

is unbounded.