M04 · optional contextual support
Read a density as probability per unit
Connect continuous probability, area, Gaussian scale and numerical integration.
Start with arithmetic, percentages and basic algebra. Open any section directly. Practice and completion here do not change any AI course requirement or the original five-lesson mathematics pilot.
1. Try a concrete case
A uniform density of 2 per metre on an interval 0.5 metres wide has probability 2×0.5=1. A density can exceed one; its area, not its height, is the probability.
2. Predict before changing a value
Change mean and spread, then increase the number of midpoint rectangles used for probability between −1 and 1. Inspect each rectangle area and distinguish numerical approximation error from uncertainty about the distribution. Name the quantity you expect to change and one you expect to stay fixed before using the controls.
3. Experiment
Change one quantity, follow every calculation
- Raw midpoint area estimate in [−1,1]
- 0.68350
- Density at mean
- 0.39894
- Variance
- 1.00000
- Uniform density 2 over width 0.5: area
- 1.00000
Vertical axis: Density per x unit. Fixed scales; 0 points clipped for display. Lines join sampled values to aid reading; they do not create extra observations. Exact numerical values are below.
| Subinterval midpoint | Density | Approximate mass |
|---|---|---|
| -0.90000 | 0.26609 | 0.05322 |
| -0.70000 | 0.31225 | 0.06245 |
| -0.50000 | 0.35207 | 0.07041 |
| -0.30000 | 0.38139 | 0.07628 |
| -0.10000 | 0.39695 | 0.07939 |
| 0.10000 | 0.39695 | 0.07939 |
| 0.30000 | 0.38139 | 0.07628 |
| 0.50000 | 0.35207 | 0.07041 |
| 0.70000 | 0.31225 | 0.06245 |
| 0.90000 | 0.26609 | 0.05322 |
A continuous density height is not a point probability. Interval probability is area; any single point has probability zero in this model. Midpoint quadrature has numerical error, distinct from uncertainty about the distribution. Differential entropy depends on units and may be negative.
Change mean and spread, then increase the number of midpoint rectangles used for probability between −1 and 1. Inspect each rectangle area and distinguish numerical approximation error from uncertainty about the distribution.
Controls save locally as practice. No timers, remote services or GPU are required. Maximum numerical work is a bounded small table or 256 bootstrap resamples.
4. Understand the calculation
Start with a calculation you can inspect
A uniform density of 2 per metre on an interval 0.5 metres wide has probability 2×0.5=1. A density can exceed one; its area, not its height, is the probability.
Connect continuous probability, area, Gaussian scale and numerical integration. The experiment changes one named quantity at a time and shows the intermediate values, rather than asking you to accept a final score. Before changing a control, write down which value you expect to move and why.
Read the formula in words and symbols
P(a≤X≤b)=∫a^b density(x) dx; Gaussian density = exp(−(x−μ)²/(2σ²))/(σ√(2π))
μ is the Gaussian mean, σ is its positive standard deviation, and σ² its variance. ∫ means accumulate small areas. Density units are inverse outcome units. The square root √ chooses the nonnegative number whose square gives its argument.
A subscript names a coordinate, case or time index; it is not multiplication. A summation sign means add the terms for the stated indices. Parentheses identify which calculation happens first. Carry units through arithmetic: adding incompatible quantities or confusing a rate with a count changes the question.
Connect the visual and the numbers
Change mean and spread, then increase the number of midpoint rectangles used for probability between −1 and 1. Inspect each rectangle area and distinguish numerical approximation error from uncertainty about the distribution. The plot uses declared fixed axes, so a new setting does not silently rescale the apparent change. Any clipped values are counted and remain available in the numerical table. Do not infer an exact curve or a new observation from a line drawn between points.
Use it in an AI experiment
These ideas support C05, C06, C10, C11, C12, C13. Use the return link when you entered from an AI lesson; its original section and experiment remain separate. A mathematical support calculation is a way to inspect an assumption, not evidence that a trained model generalizes.
Optional derivation and implementation depth
For a continuous variable, the probability at one exact point is zero even when density is positive. Log likelihood uses density values and depends on measurement units. A transformed density includes the absolute Jacobian factor; ignoring it breaks normalization, as in squashed continuous policies. Gaussian reparameterization x=μ+σε carries a fixed base-noise draw through differentiable parameters.
Recompute the worked example with a separate piece of arithmetic or a short program. State the domain and test a boundary case before using the formula elsewhere. Guided numerical practice, independent implementation and human review are different outcomes.
Optional implementation and research depth
For a continuous variable, the probability at one exact point is zero even when density is positive. Log likelihood uses density values and depends on measurement units. A transformed density includes the absolute Jacobian factor; ignoring it breaks normalization, as in squashed continuous policies. Gaussian reparameterization x=μ+σε carries a fixed base-noise draw through differentiable parameters.
Recompute one row independently, then test a boundary case and state which assumption it probes. Passing the supplied practice case does not establish independent implementation proficiency.
5. Apply it to a different case
Practice not yet passed; independent mastery and human review are not claimed.
6. Continue when useful
Completion not recorded. Skipping retains settings and practice; it never awards completion.
Used in C05, C06, C10, C11, C12, C13. Browse every support topic and course connection.