Derive no-click probability and sketch implications

Assume each impression is clicked independently with probability $p$.

1. For $A$ impressions, derive $P(\text{no clicks})$ in terms of $A$ and $p$, and explain why, when $p=0.3$, the expression is $(1-0.3)^A$ rather than $0.3^A$ if the y-axis represents $P(\text{no clicks})$. Describe qualitatively how this curve changes as $A$ grows.

2. Extend the setup to a general $p$ and compute the expected number of impressions until the first click, using the geometric distribution.

3. If click propensity differs across users according to a $\text{Beta}(\alpha,\beta)$ prior (heterogeneity), derive the marginal probability of no clicks after $A$ impressions and write it in terms of Beta functions (Beta–Binomial). Then explain how heterogeneity alters the tail relative to the i.i.d. case.

4. Suppose $p$ decreases with impression index because of fatigue, for example $p_k = p_0 e^{-\lambda(k-1)}$. Give an expression, or a tight bound, for $P(\text{no clicks after } A \text{ impressions})$ and discuss how you would estimate $\lambda$ from data.

Overview: This question tests probability and statistical modeling skills—specifically Bernoulli trial aggregation, geometric waiting-time expectations, Beta–Binomial marginalization for heterogeneous click propensities, and time-varying (exponential decay) click probabilities—relevant to Data Scientist roles in the Statistics & Math category.