Model session times and comments with exponential/Poisson

Assume each user’s probability of ending a session is memoryless (constant hazard over time). (a) Derive the implied distribution of session durations and state the memoryless property. (b) Describe two empirical checks using survival plots or hazard estimates to validate this assumption. Separately, suppose each time a user views a post they independently leave a comment with small probability p, and a user views M posts. (c) Justify a Poisson model for the per-user comment count; specify its parameter in terms of p and M and state the conditions under which the approximation is accurate. (d) Describe diagnostics that would suggest overdispersion or zero inflation and an alternative model you would consider.