Estimation and Quantitative Reasoning Questions

Sample space (S): the set of all possible outcomes of a random experiment. An event: any subset of S (one or more outcomes) to which we can assign a probability.

Example — rolling two fair six-sided dice (die1, die2):
S = {(1,1),(1,2),...,(1,6),(2,1),...,(6,6)} — 36 equally likely ordered pairs.

Define two events:

• A: "sum is 7" = {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}. |A| = 6 → P(A) = 6/36 = 1/6.
• B: "both dice are even" = {(2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6)}. |B| = 9 → P(B) = 9/36 = 1/4.

Mutually exclusive (disjoint): two events cannot occur together; A ∩ B = ∅ implies P(A ∩ B) = 0.
Independent: occurrence of one event does not change the probability of the other; equivalently P(A ∩ B) = P(A)P(B).

Compute intersection:

• A ∩ B = outcomes that are both sum 7 and both even. Sum 7 is odd, but even+even is even, so no common outcomes → P(A ∩ B) = 0.

Compare:

• P(A)P(B) = (1/6)*(1/4) = 1/24 ≠ 0 = P(A ∩ B). So A and B are not independent.
• Since P(A ∩ B) = 0, A and B are mutually exclusive.

Conclusion: For these two events, they are mutually exclusive and therefore not independent.

Assumptions of the Poisson model

• Events are independent.
• Constant average rate λ over the interval (no time-varying intensity).
• At most one event occurs in an infinitesimal interval (counts are discrete).
• Mean equals variance: E[X]=Var(X)=λ.

Given λ = 2 errors/hour.

Probability of exactly 5:
Use the Poisson PMF: P(X=k) = e^{-λ} λ^k / k!
P(X=5) = e^{-2} * 2^5 / 5! = e^{-2} * 32 / 120 ≈ 0.03609 (≈3.6%).

Probability of at least 5:
P(X ≥ 5) = 1 − P(X ≤ 4). Compute cumulative:
P(0)=e^{-2}=0.135335
P(1)=0.270671
P(2)=0.270671
P(3)=0.180447
P(4)=0.090223
Sum P(0–4)=0.947347 ⇒ P(X≥5)=1−0.947347≈0.052653 (≈5.3%).

Detecting overdispersion in real logs

• Compute sample mean μ̂ and variance σ̂^2 of counts (e.g., per hour). For Poisson σ̂^2 ≈ μ̂; overdispersion if σ̂^2 » μ̂.
• Fit a Poisson GLM and check dispersion: dispersion = residual deviance / df. Values ≫1 indicate overdispersion.
• Plot mean vs variance across groups/bins; histogram of counts vs fitted Poisson.
• Formal tests: Cameron & Trivedi dispersion test; compare Poisson vs Negative Binomial via likelihood ratio or Vuong test.

Consequences and remedies

• Overdispersion leads to underestimated SEs, too-narrow CIs, inflated Type I error — you’ll think effects are significant when they aren’t.
• Remedies: use quasi-Poisson (adjusted SE), Negative Binomial (adds variance parameter), or bootstrap CIs; model time-varying rates (e.g., Poisson process with covariates, GLM with time/season terms) or zero-inflated models if excess zeros.

Practical note: for monitoring, flagging should account for the correct variance model to avoid excessive false alarms.

Data I would collect

• Historical orders processed per day by shift and by employee (last 3–6 months) from WMS or time-study logs
• Current staffing plan: number of full-time/part-time employees, scheduled hours, planned overtime
• Non-processing time: breaks, meetings, training, expected absenteeism rate
• Order mix and complexity: average lines per order, pick/pack type, returns rate
• Operational constraints: working days next month, equipment limits, expected promotions/holidays

Simple calculation to estimate capacity

1. Compute per-employee effective throughput:
   • Effective hours = scheduled_hours − (breaks + meetings + expected_absence)
   • Throughput_per_employee_day = historical_orders_processed / average_effective_hours
2. Multiply up:
   • Estimated_monthly_capacity = Throughput_per_employee_day × effective_hours_per_day × number_of_employees × working_days
     Example:

• Throughput_per_employee_day = 50 orders/day
• Employees = 20, working_days = 22
• Estimated_monthly_capacity = 50 × 20 × 22 = 22,000 orders

Expressing uncertainty

• Provide ranges: base, conservative (−10–20% for higher absenteeism or complex orders), and optimistic (+10% if efficiency improvements apply)
• Use confidence intervals from historical variance (e.g., mean ± 1.96*std/sqrt(n)) and show scenarios (P50, P80)
• Visualize with a small dashboard: expected value, lower/upper bounds, sensitivity to key drivers (absenteeism, order complexity)

Communicating opportunity cost to operations and finance

• Translate unmet demand into dollar impact: (Orders_out_of_capacity × average_margin_per_order)
• Show trade-offs: hiring one FTE increases capacity by X orders/month at cost Y → ROI period
• Recommend short actions: temporary hires, overtime, process tweaks, or prioritization rules; quantify each option’s capacity gain and cost
• Deliver a one-page summary with assumptions, scenario numbers, recommended next steps, and risks so stakeholders can make informed operational and budget decisions.

Assumptions (explicit):

• “Initial” purchase at week0 = 100% of cohort buys at AOV.
• Week1 and week2 are repeat-purchase rates after week0. Baseline: w1=60%, w2=40%. Post: w1=63%, w2 unchanged at 40%.
• Retention after week2 decays geometrically with ratio r = w2/w1 each period (simple projection).
• No discounting, no CAC or returns; LTV = sum of expected spend over weeks 0–8.

Formula:
LTV = AOV * (1 + sum_{t=1..8} retention_t)
where retention_1 and retention_2 are given and retention_t = retention_{t-1} * r for t>2, r = w2/w1.

Baseline calculation (AOV = $50):
r = 0.40 / 0.60 = 0.6667
Retention weeks 1–8 sum ≈ 1.7298
Total purchases per user = 1 + 1.7298 = 2.7298
Baseline 8-week LTV = 2.7298 * $50 = $136.49

Post-intervention (AOV = $52, w1=0.63, w2=0.40):
r' = 0.40 / 0.63 = 0.6349
Retention weeks 1–8 sum ≈ 1.6795
Total purchases per user = 1 + 1.6795 = 2.6795
Post LTV = 2.6795 * $52 = $139.41

Result: LTV change = (139.41 − 136.49) / 136.49 ≈ +2.12% uplift.

How I’d present to Finance (brief):

• Slide 1: Key result — estimated +2.1% 8-week LTV uplift, driven by +3 ppt week1 retention and +$2 AOV.
• Slide 2: Transparent assumptions and model (show Excel with formulas), plus baseline vs post line items.
• Slide 3: Sensitivity analysis — vary week2, decay rate, and include discount rate (NPV) and margin impact. Example: if week2 also improves, upside rises materially.
• Slide 4: Statistical caveats & next steps.

Caveats / next steps:

• This is a projection based on a geometric decay assumption — validate with longer-term data or cleaner cohort A/B test.
• Check statistical significance of the w1 and AOV lifts and control for selection bias.
• Convert to economic impact: present incremental gross margin (not just revenue) and include CAC, returns, and discounting for Finance.
• Provide scenario table (conservative/expected/optimistic) in Excel for planning.

First compute net outcomes: you pay $2, so net gain X is {-2, 1, 8} with probabilities {0.6, 0.3, 0.1}.

Expected value (mean):
E[X] = (-2)(0.6) + (1)(0.3) + (8)(0.1) = -1.2 + 0.3 + 0.8 = -0.10.
Interpretation: on average the player loses $0.10 per play (the house/player offering the game gains $0.10).

Variance and standard deviation:
Compute E[X^2] = (4)(0.6) + (1)(0.3) + (64)(0.1) = 2.4 + 0.3 + 6.4 = 9.1.
Var(X) = E[X^2] − (E[X])^2 = 9.1 − (−0.1)^2 = 9.1 − 0.01 = 9.09.
Standard deviation = sqrt(9.09) ≈ 3.01.

What the metrics tell you:

• Expected value (-$0.10): average return per play; negative means the operator is profitable per play by $0.10.
• Variance (9.09) and SD (~$3.01): measure dispersion of outcomes around the mean. High SD relative to mean indicates large outcome volatility — rare big wins (+$8) drive variability even though average loss is small.

How a business stakeholder can use these numbers:

• Profitability: E[X] shows expected margin per play; multiply by expected plays to estimate revenue impact (e.g., 10,000 plays → $1,000 expected profit).
• Risk & capital planning: SD and variance indicate how often large payouts occur and how much cash reserve is needed to cover short-term variance.
• Pricing decisions: to be fair/break-even, set price = expected payout = 0.60 + 0.33 + 0.1*10 = $1.90; at $2 it's a $0.10 edge. If you want a target margin, increase price accordingly.
• Strategic use: offer as loss-leader for user acquisition if lifetime value (LTV) > cost; or limit plays/wins frequency to control tail risk.
• Statistical reliability: by law of large numbers, average loss will converge to $0.10 per play over many plays — compute confidence intervals to quantify short-term uncertainty.

Recommendation: if goal is pure immediate profit, current pricing is slightly profitable; if worried about payout volatility or customer fairness, consider lowering price volatility (cap big wins), adjust price to desired margin, or use the game for acquisition only if LTV justifies the expected loss/gain.