Estimate billboard reach and impressions

A single digital billboard stands next to a six-lane urban expressway. Estimate the weekly unique reach (the number of people who viewed it at least once) as well as total impressions, then extend the estimate to store-visit conversion with a simple Markov chain.

Data and assumptions supplied:

• Weekday traffic averages 80,000 vehicles per day and weekend traffic 50,000; each vehicle carries 1.4 people on average.
• The chance that a pass produces a viewing depends on lane and hour: $p_{vis} = 0.75$ during the day and $p_{vis} = 0.55$ at night, with 70% of traffic falling in the daytime bucket.
• Traffic splits into three segments: residents living within 3 km (30%), commuters who pass at least four weekdays a week (50%), and occasional passers-by (20%).
• Weekly pass counts follow a Poisson distribution: $\lambda = 5$ for commuters, $\lambda = 2$ for residents, and $\lambda = 1$ for occasional passers.
• People-level deduplication uses a mobile location panel of 120,000 devices per week inside a 500 m radius; each device stands for 2.2 people, and the panel capture rate carries a ±10% (1σ) uncertainty.

Tasks:

1. Work out weekly unique reach and total impressions with 95% confidence intervals, stating every formula and every independence assumption explicitly. Show the chain that turns traffic counts, occupancy, visibility and pass frequency into impressions, and the step that collapses devices to people through the panel (carry the capture-rate uncertainty through with either the delta method or a bootstrap).
2. With a three-state Markov chain (Unaware → Aware → Visit), suggest sensible transition probabilities for each segment and calculate the expected weekly visits credited to the billboard. Examine how sensitive the answer is to those probabilities and to $p_{vis}$.
3. Name at least three significant sources of bias (panel selection, deduplication error and dwell-time bias, for example) and suggest corrections or validation checks.

Overview: The item tests probabilistic modeling and statistical inference in audience measurement and attribution: Poisson frequency modeling, visibility-adjusted impressions, panel-based deduplication and expansion, uncertainty propagation through the delta method or bootstrap, and Markov-chain attribution, all within Statistics & Math / Data Science. It is used to see whether a candidate can convert traffic and visibility inputs into quantitative reach and impression figures with confidence intervals that reflect the propagated uncertainty, reason about attribution through a 3-state Markov chain, and surface measurement biases — showing both conceptual grasp and hands-on numerical estimation and sensitivity analysis.