Using the Memoryless property of Geometric RVs

Question

We're given $X_1,X_2,\ldots$ are i.i.d geometric with $$\mathbb{P}\{X_i = k\} = (0.9)^{k-1}0.1 \quad \forall \, k\in \mathbb{N}$$

Define, $$S_n = \sum_{i=1}^nX_i \\ \tau = \inf\{n \geq 1: S_n > 100\}$$

We need to find $\mathbb{E}[S_{\tau}]$.

My attempt,

I plan to use Wald's inequality, $$\mathbb{E}[S_{\tau}] = \mathbb{E}[\tau]\mathbb{E}[X_1]$$ as $\tau$ is finite (it can't go over 101) but I'm not able to get $\mathbb{E}[\tau]$.

I was asked to use the "Memoryless Property" of geometric random variables but I'm not sure how that fits with this problem at all.

Answer 1

Let $N_{\tau-1}=100-S_{\tau-1}\geq 0$. Then using memorylessness

$E(S\tau)=E(S_{\tau-1}+X_{\tau}|X_{\tau}>N_{\tau-1})=100+E(X_{\tau}-N_{\tau-1}|X_{\tau}>N_{\tau-1})=100+E(X_{\tau})$.

Answer 2

Notice that since $\tau$ is nonnegative and bounded by $101$, so $E(\tau) = \sum_{n = 1}^{101} P(\tau \geq n)$. Then notice that $\{\tau \geq n\} = \{S_{n - 1} \leq 100\}$. Also note that $S_n$ has negative binomial distribution.