Non-converting visitors before the first sale

Question

Each visitor to a store converts (buys) independently with probability $p = 0.1$ . Let $M$ be the number of non-converting visitors that arrive before the first conversion (the converting visitor is not counted).

What are the mean and variance of $M$ ?

Accepted Answer

This is the other common version of the geometric: counting failures before the first success, so the support starts at 0 (you might convert the very first visitor, giving M = 0). Its formulas: - Mean = (1-p)/p. - Variance = (1-p)/p^2 (same variance as the trials-until version). With p = 0.1: \mathbb{E}[M] = \frac{0.9}{0.1} = 9, \qquad \operatorname{Var}(M) = \frac{0.9}{0.1^2} = \frac{0.9}{0.01} = 90. Standard deviation \sqrt{90} \approx 9.49. Note this mean is exactly 1 less than the trials-until-first-success mean 1/p = 10, because that version also counts the successful trial itself. Takea…

Non-converting visitors before the first sale

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