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.
Best Answer
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 $p = P(X_0 \in S)$.