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Independent and identically distributed continuous random variables $X_1,鈥�,X_n$

Record value occurs at time $j,j鈮�n$if $X_j鈮�X_l$for all $1鈮�i鈮�j.$

Show that$E\text{[ number of record values}]=\underset{j=1}{\overset{n}{鈭�}}1/j$

Let $Y_i;i=1,2,鈥�,n$are independent and identically distributed random variables. Now define the indicator variable.

$I_j=\left\{\begin{array}{l}1鈥呪赌呪赌呪赌�\text{value recorded at time}j\\ 0鈥呪赌呪赌呪赌�\text{otherwise}\end{array}\right.$

Calculate the expected number of record values,

$E\left[\text{Number of record values}\right]=\underset{j=1}{\overset{n}{鈭�}}鈥�E\left[I_j\right]$

$=\underset{j=1}{\overset{n}{鈭�}}鈥�P\left[X_j\text{is the largest of}{X}_1,X_2,鈥�,X_j\right]$

$=\underset{j=1}{\overset{n}{鈭�}}鈥�1\left(\frac{1}{j}\right)+0\left(1鈭�\frac{1}{j}\right)$

$=\underset{j=1}{\overset{n}{鈭�}}鈥�[1/j]$

Hence, it has been shown that$E\left[\text{number of record values}\right]=\underset{j=1}{\overset{n}{鈭�}}鈥�1/j.$

Independent and identically distributed continuous random variables $X_1,...,X_n$

Record value occurs at time $j,j鈮�n$if $X_j鈮�X_1$for all $1鈮�i鈮�j$

Show that $Var$(number of record values)$=\underset{j=1}{\overset{n}{鈭�}}(j-1)/j^2$

Calculate the variance for the number of record values,

$E[Numberofrecordvalues]=$$=\underset{j=1}{\overset{n}{鈭�}}鈥�E\left[I_j\right]$

$\begin{array}{r}=\underset{j=1}{\overset{n}{鈭�}}鈥�P\left[X_j\text{is the largest of}{X}_1,X_2,鈥�,X_j\right]\end{array}$

role="math" localid="1647524674823" $=\underset{j=1}{\overset{n}{鈭�}}鈥�1\left(\frac{1}{j}\right)\left(1鈭�\frac{1}{j}\right)$

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

Hence, the required variance of the number of record value is$\underset{j=1}{\overset{n}{鈭�}}鈥�\frac{(j鈭�1)}{j^2}.$