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Let $X_1,X_2...$ be an i.i.d. continuous random variables. We say a record occurs at time $n$ and we set $Y_n=1$ if $X_n>\max\left\{ X_{1},...,X_{n}\right\} $ and $Y_n=0$ in otherwise. Let $R_n=Y_1+...+Y_n$ be the number of records that have been set by time $n$. Find $\mathbb{E}\left[R_n\right]$ and $\mathbb{\mathbb{V}\mathrm{ar}}\left[R_n\right]$.

My solution:

$\mathbb{E}\left[R_n\right]=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$

$\mathbb{\mathbb{V}\mathrm{ar}}\left[R_n\right]=1(1-1)+\frac{1}{2}\left(1-\frac{1}{2}\right)+\frac{1}{3}\left(1-\frac{1}{3}\right)+...++\frac{1}{n}\left(1-\frac{1}{n}\right)$

It's ok?

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  • $\begingroup$ Are you sure the problem is stated right? Is the inequality really strict? It seems to me that $\forall n \in \mathbb N, Y_n = 0$ a.s. $\endgroup$ – BCLC Nov 22 '15 at 17:20
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Note that by continuity we have $\omega \in \Omega$,

$$X_p(\omega) \ne X_q(\omega) \ \forall p \ne q$$

Define:

$$A_n:= (X_n > \max\{X_1, X_2, ... X_n\})$$

Observe that:

$$A_1:= (X_1 > \max\{X_1\}) = (X_1 > \max\{X_1\}) = \emptyset$$

$$A_2:= (X_2 > \max\{X_1, X_2\})$$

$$ = (X_2 > X_1) = \emptyset \ \text{if} \ (X_1 > X_2)$$

$$ = (X_2 > X_2) = \emptyset \ \text{if} \ (X_2 > X_1)$$

$$A_3:= (X_3 > \max\{X_1, X_2, X_3\})$$

$$ = (X_3 > X_1) = \emptyset \ \text{if} \ (X_1 > X_3)$$

$$ = (X_3 > X_2) = \emptyset \ \text{if} \ (X_2 > X_3)$$

$$ = (X_3 > X_3) = \emptyset \ \text{if} \ (X_3 > X_1, X_2)$$

$$\vdots$$

Thus, $$Y_n = 1_{A_n} = 1_{\emptyset} = 0$$

Hence, $$E[Y_n] = Var[Y_n] = Y_n = 0 = E[Y_n^k] \ \forall k \ge 2$$


However, if the inequality is NOT strict i.e.

$$Y_n = 1_{B_n}$$

where

$$B_n:= (X_n \ge \max\{X_1, X_2, ... X_n\}) = (X_n = \max\{X_1, X_2, ... X_n\})$$

Then you are right because

  1. $P(B_n)= 1/n$ (proof here)

  2. $B_n$'s are independent (needed in computing variance).

Consider

$B_2 = \{X_2 = \max\{X_1, X_2\}\}$

$B_3 = \{X_3 = \max\{X_1, X_2, X_3\}\}$

Since the $X_n$'s are independent, the probability that $X_3$ is the maximum among $X_1, X_2, X_3$ is independent of which is the maximum among $X_1, X_2$

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