Probability investigation: arrival dependence, route termination and incomplete evidence
Build one reproducible process-audit report with two explicitly separate experiments: counts in paired time windows and a finite routing chain. Predict exact quantities, compare bounded seeded trials with independent calculations, retain overflow and unresolved records, and defend a changed dependence/exit mechanism. These are original synthetic models, not measured service data, a queue simulation or a secure examination.
Planning estimate: 16 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
Complete the four-lesson arrival-and-absorption-processes bridge as needed: exponential-arrivals-and-counts, independent-increments-and-thinning, finite-absorption-and-tail-certificates and censored-simulation-and-dependence. The original probability lessons linked below remain canonical prerequisites; the bridge and project are directly accessible without completion locks.
Use arrival-absorption-v1 and its declared xorshift32-open-interval generator. The workbench runs 128 replicate pairs with a Poisson-process rate λ in {1,2,3,4} per hour and equal durations h in {1,2,3,4} hours. Windows are (0,h] and (h,2h]; stored arrival times are relative to their own window. Independent mode generates separate waits; dependent mode copies the first window. Each window retains at most 64 arrivals; discovering an in-window 65th arrival means count unknown with lower bound 65, not count 64. These caps are numerical resource limits, not the mathematical support of a Poisson law.
The separate routing experiment starts each of 128 paths at A and uses a listed row-stochastic rule on A,B,C,D. C and D are absorbing. Paths stop at absorption or after 64 transitions, retaining unresolved status. The regular reference has A→B 1/2,C 1/4,D 1/4 and B→A 1/3,C 1/2,D 1/6. The persistent case has Q=[[2/5,2/5],[1/5,1/2]], r_D=(0,3/10). The counterexample has A→B,C each 1/2 and B→B surely. Route steps have no physical time unit; do not infer service duration, queue capacity or completed-by-deadline output by combining these separate experiments.
Download the JSON evidence before changing controls or resetting. It retains run settings, algorithm/version, caps, generated rows and the analytical probability trace. The widget has no saved run history. Store original downloads, a manuscript, exact derivations, summary tables, figure captions and a reproduction guide separately. Each project response stores up to 8,000 characters of evidence summary and artifact references, not uploaded files and not automatically verified computation. Implementation proficiency additionally requires retained runnable code and tests; noncoding evidence can use explicit calculation tables and independently checked derivations.
A suggested protocol compares the eight arrival configurations λ,h=(1,1),(2,1),(2,2),(4,4), each in independent and copied modes. Compare all three route cases with the default arrival settings separately; do not count repeated route results as independent evidence merely because arrival controls changed. Freeze a seed schedule before viewing results. Record repeated seeds, reused rows, public-solution exposure and all deviations. A different seed is a new pseudorandom run, not a proof of independent real-world evidence.
The 16-hour estimate is an unvalidated author planning estimate. Public model solutions, timestamp labels, exported JSON and saved responses do not certify independent reasoning or reviewer identity. A real reviewer must inspect artifacts and pose a changed question; otherwise record self-review and leave independent review pending. This project closes a bounded supported investigation workflow, not the complete probability syllabus or external learner-effectiveness requirements.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
Predict, simulate, then audit
Two separate synthetic experiments: arrival counts in equal windows and a routing chain starting at A. These do not model a queue or completion deadlines. Settings and results are temporary; copy evidence into your investigation report.
Before running, predict the chance of two empty windows and whether every routing path eventually exits. Compare an independent-increments model with a process that copies its first window.
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0 of 7 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Specify the two experiments and freeze predictions
Write the question and protocol before generating the comparison runs. Define the two time windows, rate and duration units, independent versus copied mechanism, route states and transition clock. Freeze the configuration table, seed schedule, quantities to compare and treatment of caps. Predict which marginal and joint quantities distinguish the arrival mechanisms, and which route cases have finite mean absorption time. Record any prior viewing of this packet or its solutions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive the arrival and dependence references
Derive the exponential inverse sampler, the second-arrival density and the joint independent-increment mass on two windows. Derive the copied-window joint-zero probability, covariance and total variance independently of the simulation. Show one numerical case with declared rate/time units and one case with the same λh but different waiting-time scale. Explain what assumptions are built into the theoretical references.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove termination and check the finite route ledger
For both regular route cases, write the complete transition rows, transient block Q and direct-D vector. Prove a geometric survival bound before solving the first-step systems for eventual D probability and expected transitions. Independently verify the reference answers by substitution and by enumerating all outcomes through two steps. Analyze the closed-B counterexample without pretending its singular matrix has the same inverse.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Retain runs, capped paths and independent numerical checks
Execute the frozen protocol, download each JSON record before changing controls and make a manifest keyed by settings, seed and version. Recompute at least one window count from retained arrival times, one zero-pair frequency from all 128 rows and one route endpoint/transition count from its retained trace. Audit overflow and unresolved status, shared denominators and any repeated streams. For implementation, retain a second independently written checker with tests; for noncoding, retain equivalent hand-audited rows and explicit tables.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Compare predictions while separating four error sources
Compare the observed joint-zero frequencies with both models and the route results with the exact reference. Report Monte Carlo uncertainty, cap/censoring bounds, numerical rounding and model limitations separately. Explain whether a zero observed discrepancy or a small standard error could establish real-world Poisson arrivals. Retain at least one failed prediction or inconclusive case and propose a revision with a new declared check.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Assemble one reproducible investigation
Build one manuscript connecting your original predictions, arrival-law proof, route certificate, downloaded records, independent checks, comparisons and revision. Include a compact result table, an interpretable paired-window figure or trace, and a reproduction guide. State which conclusions concern the synthetic model, which concern your implementation and which would require new real data. Identify retained artifacts in the text fields without pretending they were uploaded or reviewed automatically.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Defend a changed assumption and preserve the review
Ask a reviewer to reproduce one probability, inspect one exported row and challenge one model assumption. Defend why identical one-window distributions do not imply independent increments, why the cap does not convert an unresolved route into failure, and why numerical agreement is not observational validation. Record actual questions, answers and revision-specific changes. If no independent reviewer is available, mark a self-defense and leave independent review pending.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
Targeted lesson returns
Specify the two experiments and freeze predictions
