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Find the expected number of the individuals $1$...,n who did not recruit anyone else.

Let Xi=$1$If individual i don't recruit anyone

=$0$otherwise

$\mathrm{E}\left[{\mathrm{X}}_{\mathrm{i}}\right]=P\{\mathrm{i}\text{doesn't recruit any of}\mathrm{i}+1,\mathrm{i}+2,鈥�\mathrm{n}\}$

$=\left(\frac{i鈭�1}{i}\right)\left(\frac{i}{i+1}\right)鈥�\left(\frac{n鈭�2}{n鈭�1}\right)$

$鈬�\mathrm{E}\left[{\mathrm{X}}_i\right]=\frac{i鈭�1}{n鈭�1}$

$鈬�\mathrm{E}\left[\underset{i=1}{\overset{\mathrm{n}}{鈭�}}{\mathrm{X}}_{\mathrm{i}}\right]=\frac{\underset{i=1}{\overset{n}{鈭�}}(i鈭�1)}{(n鈭�1)}$

$=\frac{n}{2}$

$\mathrm{Var}鈦�\left({\mathrm{X}}_i\right)=\frac{(i鈭�1)}{(n鈭�1)}[1鈭�\frac{(i鈭�1)}{(n鈭�1)}]$

$=\frac{(i鈭�1)(n鈭�1)}{(n鈭�1{)}^2}$

Find the expected number of the individuals $1$...,n who did not recruit anyone else is$=\frac{(i鈭�1)(n鈭�1)}{(n鈭�1{)}^2}$

Derive an expression for the variance of the number of individuals who did not recruit anyone else, and evaluate it for n=$5$.

$\text{For}i<j,\mathrm{E}\left[{\mathrm{X}}_i{\mathrm{X}}_j\right]=\frac{i鈭�1}{i}鈥�鈥�\frac{j鈭�2}{j鈭�1}\frac{j鈭�2}{j}\frac{j鈭�1}{j+1},鈥�+\frac{n鈭�3}{n鈭�1}$

$=\frac{(i鈭�1)(j鈭�2)}{(n鈭�2)(n鈭�1)}$

$鈬�\mathrm{Cov}鈦�[X_i,X_j]=\frac{(i鈭�1)(j鈭�2)}{(n鈭�2)(n鈭�1)}鈭�\frac{(i鈭�1)(j鈭�1)}{(n鈭�1)(n鈭�1)}$

$=\frac{(i鈭�1)(j鈭�n)}{(n鈭�2)(n鈭�1{)}^2}$

$鈬�\mathrm{Var}鈦�\left(\underset{i=1}{\overset{n}{鈭�}}{X}_i\right)=\underset{i=1}{\overset{n}{鈭�}}\mathrm{Var}鈦�\left(X_i\right)+2\underset{i=1}{\overset{n鈭�1}{鈭�}}\underset{j=i+1}{\overset{n}{鈭�}}\mathrm{Cov}鈦�(X_i,X_j)$

$=\frac{\underset{i=1}{\overset{n}{鈭�}}(i鈭�1)(n鈭�i)}{(n鈭�1{)}^2}+2\underset{i=1}{\overset{n鈭�1}{鈭�}}\underset{j=i+1}{\overset{n}{鈭�}}\frac{(\mathrm{i}鈭�1)(j鈭�n)}{(n鈭�2)(n鈭�1{)}^2}$

$鈬�\mathrm{Var}鈦�\left(\underset{i=1}{\overset{n}{鈭�}}{X}_i\right)=\frac{1}{(n鈭�1{)}^2}\underset{i=1}{\overset{n}{鈭�}}(i鈭�1)(n鈭�i)鈭�\frac{1}{(n鈭�2)(n鈭�1{)}^2}\underset{i=1}{\overset{n鈭�1}{鈭�}}(i鈭�1)(n鈭�i)(n鈭�i鈭�1)$

Derive an expression for the variance of the numberof individuals who did not recruit anyone else is

$鈬�\mathrm{Var}鈦�\left(\underset{i=1}{\overset{n}{鈭�}}{X}_i\right)=\frac{1}{(n鈭�1{)}^2}\underset{i=1}{\overset{n}{鈭�}}(i鈭�1)(n鈭�i)鈭�\frac{1}{(n鈭�2)(n鈭�1{)}^2}\underset{i=1}{\overset{n鈭�1}{鈭�}}(i鈭�1)(n鈭�i)(n鈭�i鈭�1)$