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Glacius
ProbabilityConcept reference

Reading density curves

A probability density assigns interval probabilities through the areas beneath its curve.

2 min read1 figureSources & further reading
On this page 9 sections
  1. Overview
  2. Understand the idea
  3. Height measures concentration per unit of x
  4. Work through the relationship
  5. A sensor model may assign a probability to falling within a tolerance band
  6. Key takeaway
  7. Further questions
  8. Sources & further reading
  9. Concept connections

01Understand the idea#

For a continuous quantity, such as a measured delay, individual exact values have zero probability under a density model. Probabilities belong to intervals. The density’s height tells you how concentrated probability is per unit of the horizontal variable; area converts that concentration into probability.

The shaded interval runs from 0.50.5 to 1.51.5. Its supplied area is 0.50.5, so it contains half the probability.

Uniform density on [0,2], zero outside. Constant height 0.5. The interval [0.5,1.5] is shaded. Its supplied area is 0.5. Total area is 1.Uniform density on [0,2], zero outside. Constant height 0.5. The interval [0.5,1.5] is shaded. Its supplied area is 0.5. Total area is 1.
Figure 1Uniform density on [0,2], zero outside. Constant height 0.5. The interval [0.5,1.5] is shaded. Its supplied area is 0.5. Total area is 1.
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A density spreads probability over a line. An interval gets the area beneath the curve; the whole curve has area 11.

P(a≤X≤b)=areaP(a\le X\le b)=\text{area}

A uniform density on [0,2][0,2] has height 0.5 because its total area must be 1. The interval [0.5,1.5][0.5,1.5] has width 1, so its area is 1(0.5)=0.51(0.5)=0.5. An interval of width 0.2 has probability 0.1 under the same density. Reading the height alone ignores interval width.

Check your reasoning

Give P(1≤X≤2)P(1\le X\le 2).

0124Area = 0.25height 0.25x
Show answer and explanation
0.25

The shaded area is 0.250.25.

02Height measures concentration per unit of x#

Height measures concentration per unit of xx. It can exceed 11. A narrow density with height 22 can still have total area 11.

At a single exact point, the width is zero and the probability is 00. Including an endpoint does not change an interval probability.

Check your reasoning

A learner uses height 22 for P(0.1≤X≤0.4)P(0.1\le X\le 0.4). Repair it.

00.10.40.5Area = 0.6height 2x
Show answer and explanation
0.6

Use the area, 0.60.6.

03Work through the relationship#

For a continuous density, the probability outside an interval is what remains after its area is removed from 11.

P(outside)=1−P(inside)\begin{gathered}P(\text{outside})\\=1-P(\text{inside})\end{gathered}
Check your reasoning

XX is a delay. Give the chance outside 0≤X≤20\le X\le 2.

025Area = 0.16height 0.4x
Show answer and explanation
0.84

1−0.16=0.841-0.16=0.84.

04A sensor model may assign a probability to falling within a tolerance band#

A sensor model may assign a probability to falling within a tolerance band. That is an area, not the density at the target reading. Changing measurement units changes density heights while leaving probabilities of corresponding intervals unchanged.

Key takeaway

Use the supplied area for an interval probability. Heights are concentrations per unit, and total area is one.

  • Interpret a supplied area under a continuous density curve as an interval probability.

Further questions

How do a normal density’s center and spread describe its shape?
The normal mean locates the center. Its positive standard deviation controls the horizontal scale; interval probabilities still come from areas.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Reading density curves.” Math behind ML. /learn/p-density-meaning