Q.7.13 - A First Course in Probability

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

Step-by-Step Solution

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

Step 2: 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,

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

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

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

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

Step 3: Final Answer (Part a)

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

Step 1: 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)$=\sum _{j=1}^n(j-1)/j^2$

Step 2: Explanation (Part b)

Calculate the variance for the number of record values, 

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

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

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

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

Step 3: Final Answer (Part b)

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