AI Grounds
Open AI Grounds on a desktop
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AI Grounds
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
Guided discovery
Connect density area to cumulative probability.
f(x) = 0.25 on [0, 4]; density is zero outside that interval. Each model has total density area 1. Values of x use an arbitrary unit; density is measured per x-unit.
Cannot exceed Upper bound. Exact editor or keyboard slider; steps of 0.05. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Cannot fall below Lower bound. To shift right, raise this bound first. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Solid line · density f(x). Shaded region · selected interval probability. Dashed lines mark the selected bounds.
The fixed density scale is 0–2.25 across all models. Density height can exceed 1; shaded area is probability and stays within 0–1.
Area under the density in these bounds.
0.2500
F(upper) − F(lower)
Probability at or below the lower bound.
0.2500
P(X ≤ 1)
Probability at or below the upper bound.
0.5000
P(X ≤ 2)
Probability = F(2) − F(1) = 0.5000 − 0.2500 = 0.2500.
| Bound x | Density f(x) | Cumulative F(x) |
|---|---|---|
| Lower: 1 | 0.2500 | 0.2500 |
| Upper: 2 | 0.2500 | 0.5000 |
Within the model’s values: width 1.0000 × average density (0.2500 + 0.2500) / 2 = area 0.2500.
The selected width inside the model is clipped to its supported values. These flat or linearly rising densities give exact rectangle or trapezoid areas. A collapsed or entirely outside interval has area 0.
A probability density function (PDF), written f(x), is nonnegative and has total area 1. The cumulative distribution function (CDF), written F(x), equals P(X ≤ x). For each supplied continuous density model, the probability of [lower, upper] is F(upper) − F(lower). Every single point has probability zero; including or excluding endpoints changes no interval probability. This does not apply to discrete point masses.
Uniform 0–4: F(x) = x/4 within [0, 4]. Rising triangle: F(x) = x²/4 within [0, 2]. Narrow uniform: F(x) = 2x within [0, 0.5]. Each CDF is 0 below its interval and 1 above. Endpoint PDF values use the displayed closed-interval formula; changing the value at a single endpoint would not change areas.
These are exact teaching distributions. A CDF curve may be shown by closely spaced line segments, but endpoint values and rectangle/trapezoid probabilities are calculated directly from the exact formulas.
NIST · Uniform distribution