Continuous Uniform Distribution: Equal Likelihood Over an Interval

The probabilistic experiment behind the continuous uniform distribution consists of selecting a value at random from a fixed interval, where no value within the interval is preferred over another. Any subinterval of equal length has the same probability of being selected. The randomness lies purely in position within the interval, not in frequency or accumulation of events.

The Probabilistic Experiment Behind continuous uniform distribution

The probabilistic experiment behind the continuous uniform distribution begins with the assumption that an outcome can occur anywhere within a fixed interval, and that no location inside that interval is preferred over another. The key idea is not randomness over trials, but randomness over position or value. Once the bounds of the interval are set, the experiment treats every point between them as equally plausible.

This type of experiment arises when the only information available is that a value lies somewhere between two limits, and there is no mechanism that biases one sub-interval over another. The uncertainty is purely geometric: probability corresponds to relative length, not to isolated points. Because the outcomes form a continuum, individual values have zero probability; only ranges matter.

The defining feature of this experiment is complete symmetry across the interval. If one interval is twice as long as another, it is twice as likely to contain the outcome. Nothing else distinguishes outcomes.

Like the discrete uniform distribution, the continuous uniform distribution is built on the idea of complete symmetry, no outcome is favored over another. The difference lies only in how probability is assigned: discrete uniform spreads probability across a finite set of distinct values, while continuous uniform spreads it evenly across an entire interval, with probability determined by length rather than individual points.

Example:

Choose a real number at random between 0 and 10, with no additional information. The chance that the number lies between 2 and 4 depends only on the interval length (2 units), not on where it sits inside the range.

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Notation

X \sim U(a, b)

— distribution of the random variable.

X \sim \text{Uniform}(a, b)

— alternative explicit form.

U(a, b)

\text{Unif}(a, b)

— used to denote the distribution itself (not the random variable).

U(0, 1)

— the standard uniform distribution on the unit interval.

Note: The continuous uniform distribution is distinct from the discrete uniform distribution. The continuous version has a probability density function, while the discrete version has a probability mass function.

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Parameters

a: lower bound of the interval, where

a \in \mathbb{R}

b: upper bound of the interval, where

b \in \mathbb{R}

and

b > a

The continuous uniform distribution is fully characterized by these two parameters. a and b define the endpoints of the interval where the random variable can take values. The distribution assigns equal probability density to every point in this interval, making it the simplest continuous distribution.

Probability Density Function (PDF) and Support (Range)

Continuous Uniform Distribution

Constant probability over an interval

Lower Bound (a): 0

Upper Bound (b): 10

Explanation

The continuous uniform distribution has constant probability density over the interval $[a, b]$ . The probability density function is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$ , and $0$ otherwise. The expected value is $E[X] = \frac{a+b}{2}$ and the variance is $\text{Var}(X) = \frac{(b-a)^2}{12}$ . This distribution models situations where all values in an interval are equally likely, such as the position of a randomly thrown dart on a board, random arrival times within a time window, or measurement errors uniformly distributed within tolerances.

Cumulative Distribution Function (CDF)

Continuous Uniform Distribution CDF

Visualizing probability accumulation for continuous uniform distribution

Continuous Uniform - CDF

Linear increase from 0 to 1 over [a, b]

Lower Bound (a): 0

Upper Bound (b): 10

CDF Explanation

The cumulative distribution function (CDF) of the continuous uniform distribution is

F(x) = \frac{x-a}{b-a}

for

a \leq x \leq b

F(x) = 0

for

x < a

, and

F(x) = 1

for

x > b

. The CDF shows the probability that the random variable

X

is less than or equal to

x

, i.e.,

P(X \leq x)

. For the uniform distribution, this probability increases linearly from 0 to 1 across the interval. This means that the probability of landing in the first half of the interval is exactly 0.5, and the probability increases uniformly as we move through the interval.

Continuous Uniform Distribution

Continuous Uniform Distribution: Equal Likelihood Over an Interval

The Probabilistic Experiment Behind continuous uniform distribution

Notation

Parameters

Probability Density Function (PDF) and Support (Range)

Continuous Uniform Distribution

Explanation

Cumulative Distribution Function (CDF)

Continuous Uniform Distribution CDF

Continuous Uniform - CDF

CDF Explanation

Expected Value (mean)

Variance and Standard Deviation

Mode and Median

Quantiles/Percentiles

Moment Generating Function

Real-World Examples and Common Applications

Interactive Calculator

Special Cases

Properties