Question

Let $X_1,\dots ,X_n$be independent and identically distributed continuous random variables. We say that a record value occurs at time $j,j\le n,$if $X_j\ge X_l$for all $1\le i\le j$. Show that

(a) $E\left[\text{number of record values}\right]=\sum _{j=1}^{n}1/j$

(b) $\mathrm{Var}\left(\text{number of record values}\right)=\sum _{j=1}^n (j-1)/j^2$

Short answer

a) It has been shown that$E\left[\text{number of record values}\right]=\sum _{j=1}^n 1/j$

b) It has been shown that$\mathrm{Var}\left(\text{number of record values}\right)=\sum _{j=1}^n (j-1)/j^2$

Step-by-step solution

Given Information (Part a)

Independent and identically distributed continuous random variables $X_1,\dots ,X_n$

Record value occurs at time $j,j\le n$if $X_j\ge X_l$for all $1\le i\le j.$

Show that$E\text{[ number of record values}]=\sum _{j=1}^{n}1/j$

Explanation (Part a) 

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

$I_j=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{value recorded at time}j\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{otherwise}\end{array}\right.$

Calculate the expected number of record values,

$E\left[\text{Number of record values}\right]=\sum _{j=1}^n E\left[I_j\right]$

$=\sum _{j=1}^n P\left[X_j\text{is the largest of}{X}_1,X_2,\dots ,X_j\right]$

$=\sum _{j=1}^n 1\left(\frac{1}{j}\right)+0\left(1-\frac{1}{j}\right)$

$=\sum _{j=1}^n [1/j]$

Final Answer (Part a) 

Hence, it has been shown that$E\left[\text{number of record values}\right]=\sum _{j=1}^n 1/j.$

Given Information (Part b) 

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

Record value occurs at time $j,j\le n$if $X_j\ge X_1$for all $1\le i\le j$

Show that $Var$(number of record values)$=\sum _{j=1}^n(j-1)/j^2$

Explanation (Part b)  

Calculate the variance for the number of record values,

$E[Numberofrecordvalues]=$$=\sum _{j=1}^n E\left[I_j\right]$

$\begin{array}{r}=\sum _{j=1}^n P\left[X_j\text{is the largest of}{X}_1,X_2,\dots ,X_j\right]\end{array}$

role="math" localid="1647524674823" $=\sum _{j=1}^n 1\left(\frac{1}{j}\right)\left(1-\frac{1}{j}\right)$

$=\sum _{j=1}^n \frac{(j-1)}{j^2}$

Final Answer (Part b)  

Hence, the required variance of the number of record value is$\sum _{j=1}^n \frac{(j-1)}{j^2}.$