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I'm trying to understand the next exercise:

It is known that the planets in the universe can be classified into $n$ types. Planets of each type can be considered to be equally abundant. Let $ X_n $ be the number of planets that must be visited to find at least one planet of each type. Is it clear that $ X_n $ is a discrete random variable.

My first question is Which is the sample space where $X_n$ work?

I think the experiment was Choose a collect of planets of at least $n$ planet and thus the sample space is formed for all the collects of $n$ or more planets. But I'm not sure because, if I have a collect $A$ of $n$ planets of type $n$, then how much is $X_n(A)$? It's zero?

What do you think? Any help will be welcome.

Sofía
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    This is the coupon collector's problem – Henry Aug 02 '21 at 07:12
  • Saying "$n$ planets of type $n$" is probably not what you intended to say, and I am not sure what $X_n(A)$ is supposed to mean. But you are correct that $X_n$ is a non-negative integer at least $n$, so $X_n\not=0$ unless $n=0$. – Henry Aug 02 '21 at 07:20
  • Related https://math.stackexchange.com/questions/28905/expected-time-to-roll-all-1-through-6-on-a-die and https://math.stackexchange.com/questions/379525/probability-distribution-in-the-coupon-collectors-problem – Henry Aug 02 '21 at 07:23

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If we let $\Omega$ be the set of $n$ planet types, then the domain of $X$ is a subset of an infinite Cartesian product space:

$$\chi:=\{ \omega \in \Omega^{\infty}: |\{\omega\}|=n\}$$

Then $X(\omega): \chi \to \mathbb{N}$ as follows:

$$X(\omega) = \arg \min_k : |\{(\omega)_1^k\}|=n$$

Annika
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