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Good morning,

I have been trying to understand a notation of a problem relating to record values. Here is notation:

$$ \text{Let } N = \min \{n: n \gt 1, \text{ and a record occurs at time n} \}.$$

This is a part of a problem which is stated as: Let $X_1, X_2, \dots$ be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time $n$ if $$X_n \gt \max(X_1, .\dots ,X_{n−1}).$$ That is, $X_n$ is a record if it is larger than each of $X_1, . \dots ,X_{n−1}$.

Could anyone please give me an explanation of that notation or what is the meaning of $N$?

PS I was told "The event that a record occurs at time $n$ but not before at times $2, \dots, n − 1$" but I don't understand why.

Thank you so much!

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Yes, that's what the text book says, and André Nicolas' explanation seems reasonable. Thank you very much for your effort though. :) –  Heber Sarmiento Feb 20 '13 at 22:46

1 Answer 1

up vote 0 down vote accepted

We line up a bunch of people, and find their weights. Let $X_1$ be the weight of the first person we weigh, $X_2$ be the weight of the second person we weigh, and so on. We will give our examples in integer numbers of pounds, but should remember that the $X_i$ are in principle continuously distributed. That means in particular that in this model, two different people have the same weight with probability $0$.

Suppose that the weights are, in order of weighing, $172, 143, 162, 97, 245, 330, 83, 106, \dots$. Then $245$ is the first weight (after $172$), which is bigger than all the previous ones. This occurred on the fifth weighing, so for this sequence of weighings, the random variable has taken on the value $5$.

Similarly, if the sequence of weighings results in $87,88, 400, 277, 123, \dots$, then the second weighing (after $87$) is the first one that beats all the previous ones. So in this case, the random variable $N$ has taken on the value $2$. The fact that the third person is huge doesn't matter.

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