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.

  • 2
    $\begingroup$ $\tau$ is a geometric random variable, no? $\endgroup$ – Mike Earnest Dec 28 '18 at 19:27
  • 1
    $\begingroup$ so there's no further information on $X_n$ except it being real-valued and i.i.d. ? $\endgroup$ – 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 $p = P(X_0 \in S)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.