Q.7.34 - A First Course in Probability | StudyQuestionHub

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Q.7.34

Question

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 $E [T_{r} ∣ T_{r - 1}]$ .

(b) Determine $E [T_{r}]$ in terms of $E [T_{r - 1}]$ .

(c) What is $E [T_{1}]$ ?

(d) What is $E [T_{r}]$ ?

Step-by-Step Solution

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Answer

It has been determined that $E [T_{r} ∣ T_{r - 1}] = T_{r - 1} + 1 + (1 - p) E [T_{r}]$
It has been determined that $E [T_{r}] = \frac{1}{p} + \frac{1}{p} E [T_{r - 1}]$ .
It has been found that $E [T_{1}] = \frac{1}{p}$ .
It has been found that $E [T_{r}] = \sum_{i = 1}^{r} {(\frac{1}{p})}^{i}$ .

1Step 1: Given information (Part a)

Tr denote the number of flips required to obtain a run of r consecutive heads

2Step 2: Solution (Part a)

$p =$ the probability that a coin lands on heads

$E [T_{r}] =$ the number of flips required to obtain a run of r consecutive heads.

Find $E [T_{r} ∣ T_{r - 1}]$

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

3Step 3: Final answer (Part a)

It has been determined that $E [T_{r} ∣ T_{r - 1}] = T_{r - 1} + 1 + (1 - p) E [T_{r}]$

4Step 4: Given information (Part b)

Tr denote the number of flips required to obtain a run of r consecutive heads

5Step 5: Solution (Part b)

Find $E [T_{r}]$

If expectations on both sides of (a) yields,

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

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

6Step 6: Final answer (Part b)

It has been determined that $E [T_{r}] = \frac{1}{p} + \frac{1}{p} E [T_{r - 1}]$ ̣

7Step 7: Given information (Part c)

Tr denote the number of flips required to obtain a run of r consecutive heads

8Step 8: Solution (Part c)

Find $E [T_{1}]$

Substitute 1 for r in part (b).

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

$= E [T_{0}] + 1 + (1 - p) E [T_{1}]$

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

$= \frac{1}{p}$

$[E [T_{0}] = 0]$

9Step 9: Final answer (Part c)

It has been found that $E [T_{1}] = \frac{1}{p}$

10Step 10: Given information (Part d)

Tr denote the number of flips required to obtain a run of r consecutive heads

11Step 11: Solution (Part d)

Find $E [T_{r}]$

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

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

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

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

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

$= \sum_{i = 1}^{r} {(\frac{1}{p})}^{i}$

12Step 12: Final answer (Part d)

It has been found that $E [T_{r}] = \sum_{i = 1}^{r} {(\frac{1}{p})}^{i}$

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