Derive paying users over time with churn
Overview
This question evaluates the ability to model user conversion and churn with discrete-time recurrence relations, derive closed-form expressions for payers over time, and analyze steady-state and stability conditions.
Uber
Oct 13, 2025, 9:49 PM
Data Scientist
Onsite
Statistics & Math
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Leaky-Bucket Model of Paying Users
Context
- Time is discrete by month t = 1, 2, ...
-
Each month t:
- N new users start a free trial.
- a (fraction, 0 ≤ a ≤ 1) convert to paid at month-end.
- b (fraction, 0 ≤ b ≤ 1) of existing start-of-month paying users churn at month-end.
- Let P_t be the number of paying users at the end of month t. P_0 is given.
Tasks
(i) Derive P_1, P_10, and a closed-form P_M in terms of P_0, N, a, b (assume churn does not apply to the same-month converters).
(ii) Find the steady state P* and the condition for convergence/stability.
(iii) Re-derive if churn also applies to same-month converters (i.e., churn is applied to all payers at month-end).
(iv) If N, a, b vary by month, write the general recurrence and state when P_M has a closed form.
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