Method of moments for a geometric waiting count

Question

A market maker sends quotes one at a time until one gets hit, then starts over. The number of quotes needed each round (counting the one that gets hit) behaves like a geometric random variable on $1, 2, 3, \dots$ with unknown hit probability $p$ . Across many rounds the average number of quotes per hit is $5$ .

Use the method of moments to estimate $p$ .

Accepted Answer

One unknown, one equation. The method of moments matches the model's mean to the sample average. For a geometric variable counting trials up to and including the first success, the mean is \mathbb{E}[X] = \frac{1}{p}. Solve. Set the model mean equal to the observed average of 5: \frac{1}{p} = 5 \implies \hat{p} = \frac{1}{5} = 0.2. Sanity check. A hit probability of 0.2 means about one quote in five gets hit, so on average it takes five quotes to land one. That matches the data, as it must, because we chose \hat p precisely to reproduce the observed average. Takeaway. For a geometric count th…

Method of moments for a geometric waiting count

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