Domain: Natural Science — Statistics & Data Science
Target concept: Poisson distribution

Video generation prompt: A white background displays the probability mass function of a Poisson distribution with λ = 2. Then, keeping the same scale, λ is increased by 1 successively up to 5, and the changes in the probability mass function are observed.

Evaluation guide:
(i) Target concept: Visual parametric evolution of discrete probability mass functions under the Poisson family.
Minimum mechanism: Bar charts must encode exact integer support, correct modal location (at λ for integer λ), monotonic decay from mode, and increasing dispersion, all determined solely by λ value and independent of rendering choices.
(ii)Storyline: Frame 0–2: λ=2 PMF appears, labeled, stable. Frame 2–4: Smooth transition to λ=3 PMF (same axes, same scale, new label). Frame 4–6: Transition to λ=4. Frame 6–8: Transition to λ=5. Frame 8–10: All four states are held together in a comparative layout (side by side or stacked) for a final direct comparison. Throughout: fixed overhead/frontal viewpoint; white background unbroken; no camera motion; no extraneous objects; all labels and bars fully visible and unoccluded.

Manual:
Experimental Procedure
1. Display Poisson PMF for λ=2 on white background with labeled axes and λ value.
2. Sequentially replace with PMFs for λ=3, λ=4, λ=5 — keeping identical scale and layout.
3. Maintain fixed viewpoint and side-by-side or stacked spatial arrangement for direct comparison.
Observation of Experimental Phenomena
- Each PMF is a discrete bar chart over non-negative integers.
- Mode shifts rightward: λ=2 peaks at x=2, λ=3 at x=3, etc.
- Spread increases: higher λ covers wider x-range with flatter tails.
- Heights preserve relative Poisson ratios (e.g., p(0;2) < p(1;2) ≈ p(2;2)).
Summary of Experimental Principles
Poisson distribution models count data with fixed average rate λ; PMF is p(k) = e^{-λ} λ^k / k!; mean = λ, variance = λ; as λ increases, distribution becomes more symmetric and dispersed.

Prompt Grounding rubric:
level_1: No white background is shown; no Poisson PMF bar chart appears; there is no sequence of λ values displayed; no numeric λ label appears anywhere; there is no side-by-side or sequential comparison of distributions.
level_2: A white background is present, but only a single static bar chart is shown with no λ label, or with a λ value other than 2, 3, 4, or 5; there is no change in λ over the course of the clip; no variation in bar pattern or spread is visible.
level_3: A white background is present; at least one correctly labeled Poisson PMF graph appears (λ=2); at least two of the four λ values (2, 3, 4, 5) are shown, but not all four; graphs appear sequentially or side by side, but in the wrong order (e.g., λ=4 shown before λ=2) or with a transition skipped.
level_4: A white background is present; all four λ values (2, 3, 4, 5) are visualized in ascending order; each graph is labeled with its λ value; all graphs share the same horizontal and vertical scale; the graphs are arranged so they can be directly compared (side by side or stacked, from a fixed viewpoint); one non-core flaw is present: e.g., missing x-axis ticks, faint labeling, or one graph briefly hidden during a transition.
level_5: All conditions for score 4 are met; in addition, all four λ values (2, 3, 4, 5) appear in strict ascending order (2 → 3 → 4 → 5); every graph is labeled with a bold, legible, centered λ value; all four graphs share identical axis limits, tick spacing, and bar width; the spatial arrangement stays consistent throughout (e.g., a fixed left-aligned stack or an evenly spaced row); no graph is hidden, no label is misplaced, and no chart's scale drifts from the others.

Scientific and Causal Correctness rubric:
level_1: The bar charts don't look like Poisson PMFs: they might look like smooth Gaussian curves, uniform-height bars, or exponentially decaying bars instead of discrete integer bars; the x-axis doesn't show non-negative integers 0,1,2,…; or there is no visible shift in where the bars peak or how spread out they are as λ increases from 2 to 5.
level_2: Bars are present but placed incorrectly: e.g., the x-axis starts at 1 instead of 0, skips integers, or includes negative values; at least one graph's tallest bar sits somewhere clearly wrong for its λ (e.g., the λ=2 graph peaks at x=0 or x=4 instead of near x=2); or there is no visible increase in where the bars are centered, or in how spread out they are, as λ increases.
level_3: Every graph shows bars only at non-negative integers (0,1,2,…); the tallest bar's position moves rightward as λ increases (e.g., λ=2 peaks near x=2, λ=3 near x=3); bar heights decrease moving away from the tallest bar; the bars visibly spread wider as λ increases from 2 to 5; but one clear violation remains: e.g., the λ=4 graph's tallest bar sits at x=3 instead of near x=4, or the λ=5 graph shows no bar at all at x=5.
level_4: All graphs show bars only at non-negative integers with no gaps in the visibly non-negligible range; each graph's tallest bar sits at or immediately next to its own λ value (λ=2→peak near 2, λ=3→peak near 3, λ=4→peak near 4, λ=5→peak near 5); the bars broaden and the peak flattens somewhat as λ increases; all four graphs use the same bar width and axis alignment; at most one small flaw remains, such as one bar in one graph looking slightly taller or shorter than its neighbors would suggest.
level_5: All conditions for score 4 are met; in addition, within each graph the bars fall off smoothly and monotonically on both sides of the peak with no bar unexpectedly taller than a bar closer to the peak; comparing across the four graphs, both the rightward shift of the peak and the increasing spread/flattening are unambiguous at a glance; and every bar chart shows a clean, uncluttered discrete bar chart with no smoothing, curve-fitting, or interpolation drawn over the bars.

Spatiotemporal Consistency rubric:
level_1: The white background changes color or texture partway through; a PMF graph appears or disappears with no visible transition; a graph's scale or position jumps abruptly between frames; the lighting shifts abruptly (e.g., a shadow suddenly appearing then vanishing); or a bar chart appears to break apart into disconnected pieces.
level_2: The white background stays, but it visibly flickers or pulses in brightness; a graph resizes abruptly (e.g., the λ=3 graph suddenly looks noticeably larger than the λ=2 graph with no smooth transition); the λ label jumps to a different position between frames; a graph appears to rotate or skew during a transition; or the motion between λ states looks jerky with visible skips.
level_3: The background stays a stable, uniform white; each graph stays fixed in position and orientation once it appears; transitions between λ states (fade, slide, or replacement) show no teleporting, stretching, or skewing; however, one minor spatiotemporal flaw is visible, such as two graphs briefly overlapping during a transition, or a label appearing to settle into place slightly after its graph does.
level_4: All conditions for score 3 are met; in addition, every graph keeps the same origin and axis alignment as the ones before it; transitions are smooth and continuous (fade, cross-dissolve, or a steady slide) with no overlap or visible jitter; brightness, contrast, and sharpness look the same across all four λ states; at most one small flaw remains, such as a brief dip in opacity during one fade.
level_5: All conditions for score 4 are met; in addition, the white background looks perfectly uniform throughout with no grain, gradient, or flicker; every graph occupies the same on-screen position and size across all frames with no visible drift or shimmer; transitions look smooth and symmetric with no visible jump, blur, or stutter; and overall sharpness and contrast stay visibly consistent from the first frame to the last.