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Interpreting the Y-Axis of a Probability Density

Article Quant Q&A · Author: Guillaume Paris
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Summary

The document explains what the vertical value of a probability density function represents. For a continuous random variable, probabilities over intervals are obtained by integrating the density across those intervals, so the area under the curve over a range corresponds to the probability of falling within that range.

A density value at one exact point is not itself the probability of that outcome; for a continuous variable, a single point has zero probability. The answer describes the density’s height as having mathematical rather than direct probability interpretation. Its units are inverse to the units of the measured variable, which helps preserve a dimensionless area when density is integrated over the horizontal axis. The discussion is brief and does not provide examples or address density estimation, so it serves mainly as a conceptual clarification.

Key ideas

  • Probabilities over intervals are found by integrating the density across those intervals.
  • The area under a density curve over a range represents the probability of values in that range.
  • A density’s value at one point is not the probability of that exact value.
  • The density’s units depend inversely on the units used for the random variable.

Tags

Statistics
Full text
# y-axis unity of density probability function


# y-axis unity of density probability function












What is the unity/interpretation of the y-axis of a density distribution function? The X-axis is the values of the random variable, the area is the probabilty what about the y-axis ?

## Answer by vikram (score 1, accepted)

https://quant.stackexchange.com/a/14849

The y-value is meaning-less for interpretation. As f(x) is probability distribution function and probability being defined as: $P(x<X)=\int_{-\infty}^b\!f(x)dx$. Y-value just has mathematical usage but no physical interpretation. Can refer to the link as well.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.