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I am exploring churn and lifetime modeling. From what I see on various online material, the lifetime period is defined as 1 / churn-risk.

For example, if the churn risk for a 12-month period is estimated to be .3, the lifetime is calculated to 3.3 years by 1 / .3.

Is there a mathematical proof of this way of modeling the lifetime period? Intuitively, 1 - churn risk sounds more correct (probability of not churning).

Where does the assumption that churn risk increase over time come from? Is there a distribution formula that is derived to 1 / churn risk?

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This comes from the geometric distribution. If the probability of churning in any given year is $p$ (where $p$ is constant across years, and years are independent) then the expected number of years until you churn is $1/p$.

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