Chapter 7: Q7.34 (page 361)

7.34. For another approach to Theoretical Exercise 7.33, let Tr denote the number of flips required to obtain a run of r consecutive heads. (a) Determine E[Tr|Tr−1]. (b) Determine in terms of E[Tr−1]. (c) What is E[T1]? (d) What is E[Tr]?

Short Answer

Expert verified

$= \frac{1}{p} [E [T_{0}] = 0]$

Step by step solution

Given Information

Let be the probability that a coin lands on heads. Let Tr denote the number of flips required to obtain a run of r consecutive heads.

We have to find

$E [T r | T r - 1] .$

$E [T r - 1]$

E[T1]

$E [T r]$

Explanation Of a

Let $p$ be the probability that a coin lands on heads. Let $E [T_{r}]$ denote the number of flips required to obtain a run of $r$ consecutive heads.

Determine $E [T_{r} ∣ T_{r - 1}] .$

$E [T_{r} ∣ T_{r - 1}] = T_{r - 1} + 1 + (1 - p) E [T_{r}]$

Explanation Of b

Determine $E [T_{f}]$

Taking expectations on both sides of (a) yields,

$E [T_{r}] = E [T_{r - 1}] + 1 + (1 - p) E [T_{r}]$

$= \frac{1}{p} + \frac{1}{p} E [T_{r - 1}]$

Explanation Of c

DetermineSubstitute for r in part (b)

Explanation Of d

Determine $E [T_{r}]$

$E [T_{r}] = \frac{1}{p} + \frac{1}{p} E [T_{r - 1}]$

$= \frac{1}{p} + \frac{1}{p} [\frac{1}{p} + \frac{1}{p} E [T_{r - 1}]]$

$= (\frac{1}{p}) + {(\frac{1}{p})}^{2} + {(\frac{1}{p})}^{2} E [T_{r - 2}]$

$= (\frac{1}{p}) + {(\frac{1}{p})}^{2} + {(\frac{1}{p})}^{3} + {(\frac{1}{p})}^{3} E [T_{r - 3}]$

$= \sum_{i = 1}^{p} {(\frac{1}{p})}^{i} + {(\frac{1}{p})}^{r} E [T_{0}]$

$= \sum_{i = 1}^{r} {(\frac{1}{p})}^{i} [E [T_{0}] = 0]$

Final Answer

$E [T_{r} ∣ T_{r - 1}] = T_{r - 1} + 1 + (1 - p) E [T_{r}]$

E[Tr−1]. $= \frac{1}{p} + \frac{1}{p} E [T_{r - 1}]$

E[T1]

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7.34. For another approach to Theoretical Exercise 7.33, let Tr denote the number of flips required to obtain a run of r consecutive heads. (a) Determine E[Tr|Tr−1]. (b) Determine in terms of E[Tr−1]. (c) What is E[T1]? (d) What is E[Tr]?

Short Answer

Step by step solution

Given Information

Explanation Of a

Explanation Of b

Explanation Of c

Explanation Of d

Final Answer

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