Continuous random variables

The discrete variable page treated separate, listable support values. Suppose instead that V maps each stop consonant token to its measured voice onset time in milliseconds. Two tokens might have values 18.4 and 18.41. Between them lie 18.401, 18.402, and indefinitely many other values. How should probability be represented when the modeled values fill an interval rather than occupying separate points?

Voice onset time is an acoustic interval between stop release and the onset of voicing. It can support comparisons among measured tokens under a stated protocol, but one value does not by itself identify a phonological category or explain how a listener represents the stop.

One tempting answer is to call every variable with an uncountable range continuous. That answer is too weak for our purposes: an uncountable range does not by itself guarantee that probability can be represented by a density. We use continuous random variable for the stronger, absolutely continuous case defined below.

To see why range and distribution must be separated, consider paired formant measurements. Let

\Omega\equiv\mathbb R_+^2

and equip this space with its Borel sigma-algebra. Define the identity random vector

X:\mathbb R_+^2\to\mathbb R_+^2, \qquad X(\omega)\equiv\omega.

For every Borel set B\subseteq\mathbb R_+^2,

X^{-1}(B)=B,

so X is measurable. This fact alone does not make its distribution absolutely continuous. Absolute continuity additionally requires the probability measure to admit a density with respect to two-dimensional Lebesgue measure.

More specifically, a random variable X is absolutely continuous when there is a nonnegative function f_X such that

\mathbb{P}(X\in A)=\int_A f_X(x)\,\mathrm{d}x

for every measurable set A. We use continuous random variable for this class. This definition is stronger than merely requiring an uncountable range.

For a voice onset time measure that permits negative and positive values, a simple modeled support might be

\operatorname{supp}(V)=\mathbb{R}.

A vowel duration variable might instead have support (0,\infty) because durations are positive. The appropriate interval depends on the measurement and the scientific representation.

A point and an interval are different events

Consider the point event

V=18.417.

Under a continuous model, one point receives probability zero:

\mathbb{P}(V=18.417)=0.

This statement does not make a voice onset time near 18.417 ms impossible. An interval around the point may receive positive probability:

\mathbb{P}(18<V<19)>0.

The fact that a single point has zero probability follows because an integral over a singleton is zero. Intervals may nevertheless have positive probability.

Working through measurement resolution

Suppose an instrument reports values to the nearest millisecond. A stored token may appear as 18 ms even if its underlying measured quantity before rounding was 18.37 ms. The integer in the file does not force the model to make V discrete.

We have at least two possible representations.

A continuous model treats the stored integer as a rounded observation of an interval valued quantity.
A discrete model treats the reported integer itself as the result produced by the measurement process.

The choice depends on the target. If differences smaller than one millisecond matter and rounding is part of measurement error, the continuous representation may be useful. If only the instrument’s reported categories matter, a discrete representation may be sufficient.

Simulating the distinction in base R

Code

unrounded_vot <- c(18.37, 18.62, 19.14)
stored_vot <- round(unrounded_vot)

data.frame(unrounded_vot, stored_vot)

Several interval values can map to the same stored integer. The code illustrates rounding; it does not decide which support the scientific model should use.

Comparing discrete and continuous support

question	discrete variable	continuous variable
possible values	separate and listable	commonly represented across an interval
one value	may have positive probability	has probability zero
several values	probability found by adding masses	probability assigned across a region

The distinction concerns the modeled support. It does not follow from whether a spreadsheet column is stored as an integer or decimal.

Storage precision does not determine the modeled support

The precision of values stored in a file does not determine whether the modeled support is discrete or continuous. A column with values 18, 19, and 20 may record rounded measurements of a continuous quantity. Conversely, a slider rendered as a line may permit only seven selectable responses and thus remain discrete.

To avoid the conflation, state what the observation process can produce and what distinctions the analysis aims to represent. The display format alone does not answer either question.

Check your understanding

Give one reason to model whole millisecond response times as continuous and one reason to model them as discrete.
Explain why \mathbb{P}(V=18.417)=0 does not mean that a value near 18.417 is impossible.
Which support is more plausible for a manually assigned five point prominence score? Explain from the response process.
Give one continuous linguistic measurement whose support is restricted to positive values.

A continuous support cannot be represented by a PMF over individual points. The next page represents probability by area across intervals.

--- title: "Continuous random variables" --- The [discrete variable page](discrete-random-variables.qmd) treated separate, listable support values. Suppose instead that $V$ maps each stop consonant token to its measured voice onset time in milliseconds. Two tokens might have values 18.4 and 18.41. Between them lie 18.401, 18.402, and indefinitely many other values. How should probability be represented when the modeled values fill an interval rather than occupying separate points? [Voice onset time](https://doi.org/10.1080/00437956.1964.11659830) is an acoustic interval between stop release and the onset of voicing. It can support comparisons among measured tokens under a stated protocol, but one value does not by itself identify a phonological category or explain how a listener represents the stop. One tempting answer is to call every variable with an uncountable range continuous. That answer is too weak for our purposes: an uncountable range does not by itself guarantee that probability can be represented by a density. We use *continuous random variable* for the stronger, absolutely continuous case defined below. To see why range and distribution must be separated, consider paired formant measurements. Let $$ \Omega\equiv\mathbb R_+^2 $$ and equip this space with its Borel sigma-algebra. Define the identity random vector $$ X:\mathbb R_+^2\to\mathbb R_+^2, \qquad X(\omega)\equiv\omega. $$ For every Borel set $B\subseteq\mathbb R_+^2$, $$ X^{-1}(B)=B, $$ so $X$ is measurable. This fact alone does not make its distribution absolutely continuous. Absolute continuity additionally requires the probability measure to admit a density with respect to two-dimensional Lebesgue measure. More specifically, a random variable $X$ is [**absolutely continuous**](https://stats.libretexts.org/Bookshelves/Probability_Theory/Applied_Probability_%28Pfeiffer%29/07%3A_Distribution_and_Density_Functions/7.01%3A_Distribution_and_Density_Functions) when there is a nonnegative function $f_X$ such that $$ \mathbb{P}(X\in A)=\int_A f_X(x)\,\mathrm{d}x $$ for every measurable set $A$. We use *continuous random variable* for this class. This definition is stronger than merely requiring an uncountable range. For a voice onset time measure that permits negative and positive values, a simple modeled support might be $$ \operatorname{supp}(V)=\mathbb{R}. $$ A vowel duration variable might instead have support $(0,\infty)$ because durations are positive. The appropriate interval depends on the measurement and the scientific representation. ## A point and an interval are different events Consider the point event $$ V=18.417. $$ Under a continuous model, one point receives probability zero: $$ \mathbb{P}(V=18.417)=0. $$ This statement does not make a voice onset time near 18.417 ms impossible. An interval around the point may receive positive probability: $$ \mathbb{P}(18<V<19)>0. $$ The fact that a single point has zero probability follows because an integral over a singleton is zero. Intervals may nevertheless have positive probability. ## Working through measurement resolution Suppose an instrument reports values to the nearest millisecond. A stored token may appear as 18 ms even if its underlying measured quantity before rounding was 18.37 ms. The integer in the file does not force the model to make $V$ discrete. We have at least two possible representations. 1. A continuous model treats the stored integer as a rounded observation of an interval valued quantity. 2. A discrete model treats the reported integer itself as the result produced by the measurement process. The choice depends on the target. If differences smaller than one millisecond matter and rounding is part of measurement error, the continuous representation may be useful. If only the instrument's reported categories matter, a discrete representation may be sufficient. ## Simulating the distinction in base R ```{r} #| eval: false unrounded_vot <- c(18.37, 18.62, 19.14) stored_vot <- round(unrounded_vot) data.frame(unrounded_vot, stored_vot) ``` Several interval values can map to the same stored integer. The code illustrates rounding; it does not decide which support the scientific model should use. ## Comparing discrete and continuous support | question | discrete variable | continuous variable | |---|---|---| | possible values | separate and listable | commonly represented across an interval | | one value | may have positive probability | has probability zero | | several values | probability found by adding masses | probability assigned across a region | The distinction concerns the modeled support. It does not follow from whether a spreadsheet column is stored as an integer or decimal. ## Storage precision does not determine the modeled support The precision of values stored in a file does not determine whether the modeled support is discrete or continuous. A column with values `18`, `19`, and `20` may record rounded measurements of a continuous quantity. Conversely, a slider rendered as a line may permit only seven selectable responses and thus remain discrete. To avoid the conflation, state what the observation process can produce and what distinctions the analysis aims to represent. The display format alone does not answer either question. ## Check your understanding 1. Give one reason to model whole millisecond response times as continuous and one reason to model them as discrete. 2. Explain why $\mathbb{P}(V=18.417)=0$ does not mean that a value near 18.417 is impossible. 3. Which support is more plausible for a manually assigned five point prominence score? Explain from the response process. 4. Give one continuous linguistic measurement whose support is restricted to positive values. A continuous support cannot be represented by a PMF over individual points. The [next page](probability-density-functions.qmd) represents probability by area across intervals.