91影视

Flip the coin until a head appears.

Number of flips needed $=N$

$P\{N鈮�I\},i鈮�1=?$

We have,

$鈭�P[N=i]={鈭�}_0^1鈥�\left(\begin{array}{l}i鈭�1\\ 0\end{array}\right)p^{\mathrm{鈥�}}(1鈭�p{)}^{i鈭�1}dp$

$\begin{array}{r}={鈭�}_0^1鈥�\mathrm{p}(1鈭�\mathrm{p}{)}^{\mathrm{i}鈭�1}\mathrm{dp}\end{array}$

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

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

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

$\begin{array}{r}鈭�\mathrm{P}[\mathrm{N}鈮�\mathrm{i}]=\underset{x=i}{\overset{\mathrm{鈭�}}{鈭�}}鈥�\frac{1}{x(x+1)}\end{array}$

localid="1647429510678" $=\underset{x=i}{\overset{\mathrm{鈭�}}{鈭�}}鈥�\left[\frac{1}{x}鈭�\frac{1}{x+1}\right]$

$=\frac{1}{i}$

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

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

Flip the coin until a head appears.

Number of flips needed$=N$

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

We have,

$\begin{array}{r}\mathrm{P}[\mathrm{N}=\mathrm{i}]=\mathrm{P}[\mathrm{N}鈮�i]鈭�\mathrm{P}[\mathrm{N}>i]\end{array}$

$\begin{array}{r}=\mathrm{P}[\mathrm{N}鈮�i]鈭�\mathrm{P}[\mathrm{N}鈮�i+1]\end{array}$

$=\frac{1}{i}鈭�\frac{1}{i+1}$

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

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

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

Flip the coin until a head appears.

Number of flips needed $=N$

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

$\begin{array}{r}\mathrm{E}\left(\mathrm{N}\right)=\underset{i=0}{\overset{\mathrm{鈭�}}{鈭�}}鈥�\mathrm{P}[\mathrm{N}鈮�i]\end{array}$

role="math" localid="1647431080445" $=\underset{i=0}{\overset{\mathrm{鈭�}}{鈭�}}鈥�\left(\frac{1}{i}\right)$

$=\mathrm{鈭�}$

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