The derivation part was fine, exponential distribution, constant hazard, I've seen it before.
Start by deriving the exponential distribution from the memoryless property, then explain the property intuitively and mathematically. For empirical checks, describe using survival plots (log-scale for exponential) and hazard estimates (Nelson-Aalen) to assess fit.
Pro tip: Mention that while the exponential is the only continuous memoryless distribution, real user sessions often show decreasing hazard (e.g., due to heterogeneity), so checking for overdispersion or mixtures is crucial.
Let T be session duration. Memoryless means P(T > s+t | T > s) = P(T > t). This implies the survival function S(t) = P(T > t) satisfies S(s+t) = S(s)S(t). The only continuous solution is S(t) = e^{-λt}, so T ~ Exponential(λ).
The property means the remaining time until session end is independent of how long the session has already lasted. For example, if a user has been active for 10 minutes, the probability they stay another 5 minutes is the same as for a new user.
Plot the empirical survival function (Kaplan-Meier) on a log scale. For exponential data, this should be approximately linear. Deviations (e.g., curvature) indicate non-exponential behavior.
Estimate the hazard function using Nelson-Aalen or kernel smoothing. For exponential data, the hazard should be roughly constant over time. A decreasing hazard suggests heterogeneity or a mixture of exponentials.
If data deviates, consider alternative models like Weibull or log-normal. In A/B testing, assuming exponential when false can bias metrics like average session duration.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by framing the problem as a sum of independent Bernoulli trials and invoke the Poisson limit theorem to justify the Poisson model. Then derive the parameter λ = M * p and discuss how to check for overdispersion and zero inflation using diagnostic plots and tests. Finally, propose alternative models like negative binomial or zero-inflated Poisson if the diagnostics indicate issues.
Pro tip: Mention that in practice, p may vary across users (e.g., due to user heterogeneity), which naturally leads to overdispersion; this shows you understand real-world data complexities beyond the textbook Poisson assumption.
Explain that the total comment count is a sum of M independent Bernoulli(p) random variables. By the Poisson limit theorem, when M is large and p is small, this sum converges to a Poisson distribution with parameter λ = M * p.
Clearly state that the Poisson parameter is λ = M * p, representing the expected number of comments. Emphasize that this holds under the assumptions of independence and constant p.
Describe diagnostics: overdispersion is indicated when the variance significantly exceeds the mean (e.g., variance-to-mean ratio > 1, or a dispersion test). Zero inflation is indicated when the observed frequency of zeros is much higher than predicted by the Poisson model (e.g., a zero-inflation test or comparing observed vs. expected zeros).
If overdispersion is present, suggest a negative binomial model. If zero inflation is present, suggest a zero-inflated Poisson (ZIP) or zero-inflated negative binomial (ZINB) model. Mention that these models account for the extra variability or excess zeros.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.