Q7.34 - A First Course in Probability

Question

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]? 

Step-by-Step Solution

Answer

$=\frac{1}{p} \left[E\left[T_0\right]=0\right]$

Step 1 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  

$E\left[Tr\right|Tr-1].$

$E[Tr-1]$

 E[T1]

$E\left[Tr\right]$

Step 2 Explanation Of a

Let$p$ be the probability that a coin lands on heads. Let $E\left[T_r\right]$ denote the number of flips required to obtain a run of$r$ consecutive heads.

Determine $E\left[T_r\mid T_{r-1}\right]\text{. }$

$E\left[T_r\mid T_{r-1}\right]=T_{r-1}+1+(1-p)E\left[T_r\right]$

Step 3 Explanation Of b

Determine $E\left[T_f\right]$

Taking expectations on both sides of (a) yields,

$E\left[T_r\right]=E\left[T_{r-1}\right]+1+(1-p)E\left[T_r\right]$

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

Step 4 Explanation Of c

Determine Substitute for r in part (b)

Step 5 Explanation Of d

Determine $E\left[T_r\right]$

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

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

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

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

$=\sum _{i=1}^p{\left(\frac{1}{p}\right)}^i+{\left(\frac{1}{p}\right)}^{r}E\left[T_0\right]$

$=\sum _{i=1}^r{\left(\frac{1}{p}\right)}^i \left[E\left[T_0\right]=0\right]$ 

Step 6 Final Answer

$E\left[T_r\mid T_{r-1}\right]=T_{r-1}+1+(1-p)E\left[T_r\right]$

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

E[T1]