# The distribution of a stopping time

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.

• $\tau$ is a geometric random variable, no? – Mike Earnest Dec 28 '18 at 19:27
• so there's no further information on $X_n$ except it being real-valued and i.i.d. ? – Hayk Dec 28 '18 at 20:12

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