Question

Suppose that in Problem $7.70$, we continue to flip the coin until a head appears. Let $N$denote the number of flips needed. Find

(a)$P\{N\ge i\},i\ge 1$

(b)$P\{N=i\};$

(c)$E\left[N\right]$

Short answer

a) The value of $P\{N\ge i\},i\ge 1$is $\mathrm{P}[\mathrm{N}\ge i]=\frac{1}{\mathrm{i}};i=0,1,2,\dots .,n$

b) The value of $P\{N=i\}$is $\mathrm{P}[\mathrm{N}=\mathrm{i}]=\frac{1}{\left(i\right)(i+1)};i=0,1,2,\dots \dots ,n$

c) The value of$E\left[N\right]$is$\mathrm{\infty }.$

Step-by-step solution

Given Information (Part a)

Flip the coin until a head appears.

Number of flips needed $=N$

$P\{N\ge I\},i\ge 1=?$

Explanation (Part a) 

We have,

$\therefore P[N=i]={\int }_0^1 \left(\begin{array}{l}i-1\\ 0\end{array}\right)p^{\mathrm{\prime }}(1-p{)}^{i-1}dp$

$\begin{array}{r}={\int }_0^1 \mathrm{p}(1-\mathrm{p}{)}^{\mathrm{i}-1}\mathrm{dp}\end{array}$

localid="1647430523751" $=\frac{\left(1\right)!(i-1)!}{(2+i-1)!}$

localid="1647429441069" $=\frac{(i-1)!}{(i+1)!}$

$=\frac{1}{i(i+1)}$

$\begin{array}{r}\therefore \mathrm{P}[\mathrm{N}\ge \mathrm{i}]=\sum _{x=i}^{\mathrm{\infty }} \frac{1}{x(x+1)}\end{array}$

localid="1647429510678" $=\sum _{x=i}^{\mathrm{\infty }} \left[\frac{1}{x}-\frac{1}{x+1}\right]$

$=\frac{1}{i}$

Hence$\mathrm{P}[\mathrm{N}\ge i]=\frac{1}{\mathrm{i}};i=0,1,2,\dots .,n$

Final Answer

Hence, the value of$P\{N\ge i\},i\ge 1\text{is}\mathrm{P}[\mathrm{N}\ge i]=\frac{1}{i};i=0,1,2,\dots ,n$

Given Information (Part b)

Flip the coin until a head appears.

Number of flips needed$=N$

$P\{N=i\}=?$

Explanation (Part b) 

We have,

$\begin{array}{r}\mathrm{P}[\mathrm{N}=\mathrm{i}]=\mathrm{P}[\mathrm{N}\ge i]-\mathrm{P}[\mathrm{N}>i]\end{array}$

$\begin{array}{r}=\mathrm{P}[\mathrm{N}\ge i]-\mathrm{P}[\mathrm{N}\ge i+1]\end{array}$

$=\frac{1}{i}-\frac{1}{i+1}$

localid="1647429927042" $=\frac{1}{i(i+1)}$

$\therefore \mathrm{P}[\mathrm{N}=\mathrm{i}]=\frac{1}{\left(i\right)(i+1)};i=0,1,2,\dots .,n$

Final Answer (Part b) 

Hence, the value of $E\left[N\right]$is$\infty .$

Given Information (Part c)

Flip the coin until a head appears.

Number of flips needed $=N$

The Value of$E\left[N\right]=?$

Explanation (Part c) 

$\begin{array}{r}\mathrm{E}\left(\mathrm{N}\right)=\sum _{i=0}^{\mathrm{\infty }} \mathrm{P}[\mathrm{N}\ge i]\end{array}$

role="math" localid="1647431080445" $=\sum _{i=0}^{\mathrm{\infty }} \left(\frac{1}{i}\right)$

$=\mathrm{\infty }$

Final Answer (Part c) 

Therefore, the value of$E\left[N\right]=\infty$