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Let $(X_n)_{n\geq0}$ be a sequence of real $i.i.d$ random variables and $\tau = \inf\{n\geq0 : X_n\in S\}$ with $S \in \mathcal{B}(\mathbb{R}) $

I am trying to find $\tau$'s distribution.

Obviously, $\tau$ is a stopping time in regards to the natural filtration $\sigma(X_0,...,X_n)$ but that's all I could come up with.

Any help would be greatly appreciated.

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    $\begingroup$ $\tau$ is a geometric random variable, no? $\endgroup$ Commented Dec 28, 2018 at 19:27
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    $\begingroup$ so there's no further information on $X_n$ except it being real-valued and i.i.d. ? $\endgroup$
    – Hayk
    Commented Dec 28, 2018 at 20:12

1 Answer 1

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As Mike pointed out, $\tau$ is a geometric random variable. To see this, let $n \geq 0$ then $$\begin{align}P(\tau = n) &= P(X_0 \notin S, X_1 \notin S, \ldots, X_{n-1} \notin S, X_n \in S) \\ &= P(X_0 \notin S)P(X_1 \notin S) \times \cdots \times P(X_{n-1} \notin S) \, P(X_n \in S) \\ &= (1-P(X_0 \in S))^{n-1} \, P(X_0 \in S)\end{align}$$ which proves that $\tau$ is a geometric random variable with parameter $p = P(X_0 \in S)$.

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