Model session times and comments with exponential/Poisson

Assume the probability that a user ends a session is memoryless, meaning the hazard is constant over time. (a) Derive the distribution of session durations this implies, and state the memoryless property. (b) Describe two empirical checks based on survival plots or hazard estimates to validate this assumption. In a separate scenario, assume that on each post view a user independently leaves a comment with small probability $p$, and that a user views $M$ posts. (c) Explain why a Poisson model is appropriate for the number of comments per user; give its parameter in terms of $p$ and $M$, and state the conditions under which the approximation is accurate. (d) Describe diagnostics that would point to overdispersion or zero inflation, and an alternative model you would consider.

Overview: This question assesses knowledge of memoryless processes and count-data modeling, including exponential (constant-hazard) session-duration distributions and Poisson approximations for comment counts, within the survival-analysis and count-data-modeling topics of Statistics & Math for a Data Scientist role.

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