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Glacius
ProbabilityConcept reference

Reading probabilities from a CDF

For a<b, P(a<X≤b) equals F(b)−F(a).

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Understand the idea
  3. The lower endpoint is removed because F(a) includes X=a
  4. A closer look
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Understand the idea#

An interval probability can be the part left after one cumulative event is removed from another. For a<ba<b, subtract all values at or below aa from all values at or below bb.

P(a<X≤b)=F(b)−F(a)\begin{gathered}P(a<X\le b)\\=F(b)-F(a)\end{gathered}

If F(1)=0.2F(1)=0.2 and F(4)=0.8F(4)=0.8, the probability above 11 and at most 44 is 0.8−0.2=0.60.8-0.2=0.6.

Only supplied cumulative values F(1)=.2 and F(4)=.8 are plotted. Their vertical difference .6 equals P(1<X≤4). No CDF shape between the thresholds is invented.Only supplied cumulative values F(1)=.2 and F(4)=.8 are plotted. Their vertical difference .6 equals P(1<X≤4). No CDF shape between the thresholds is invented.
Figure 1Only supplied cumulative values F(1)=.2 and F(4)=.8 are plotted. Their vertical difference .6 equals P(1<X≤4). No CDF shape between the thresholds is invented.
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Check your reasoning

F(−1)=0.2F(-1)=0.2 and F(5)=0.7F(5)=0.7. Find P(−1<X≤5)P(-1<X\le 5).

Show answer and explanation
0.5

Subtract the nested totals: 0.7−0.2=0.50.7-0.2=0.5. The lower endpoint is excluded.

02The lower endpoint is removed because F(a) includes X=a#

The lower endpoint is removed because F(a)F(a) includes X=aX=a. The upper endpoint stays because F(b)F(b) includes X=bX=b.

Translate the interval wording before subtracting. “Above” is strict; “at most” includes equality.

Check your reasoning

A score has F(5)=0.15F(5)=0.15 and F(9)=0.65F(9)=0.65. Find the probability it is above 55 and at most 99.

Show answer and explanation
0.5

Subtract the nested totals: 0.65−0.15=0.50.65-0.15=0.5. The lower endpoint is excluded.

03A closer look#

Adding P(X=a)P(X=a) back would answer a different question, one that includes the lower endpoint.

If the two CDF values are equal, there is no probability in this band. Their difference is zero even when the endpoints are different.

Check your reasoning

F(2)=0.3F(2)=0.3, F(5)=0.8F(5)=0.8, P(X=2)=0.1P(X=2)=0.1. A draft adds the lower point mass to the CDF difference. Repair P(2<X≤5)P(2<X\le 5).

Show answer and explanation
0.5

Subtract the nested totals: 0.8−0.3=0.50.8-0.3=0.5. The lower endpoint is excluded.

Key takeaway

Subtract the lower CDF from the upper; exclude the lower endpoint and include the upper.

  • Compute P(a < X ≤ b) by subtracting supplied CDF values at the endpoints.

Sources & further reading

  1. [1]

Reference this concept

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

Glacius. “Reading probabilities from a CDF.” Math behind ML. /learn/p-cdf-interval