Continuous Uniform Distribution Calculator

Calculator Inputs

Lower bound a

Minimum possible value of the distribution.

Upper bound b

Maximum possible value of the distribution.

Point x

Used for single-point PDF and CDF evaluation.

Interval start x1

First value for interval probability.

Interval end x2

Second value for interval probability.

Quantile probability p

Choose a probability between 0 and 1.

Graph points

Controls plotted resolution for the graph.

Distribution Graph

The graph shows both the density function and cumulative distribution.

Example Data Table

This example uses default values when no submission exists.

x value	PDF f(x)	CDF F(x)
2.000	0.125	0.000
4.000	0.125	0.250
6.000	0.125	0.500
8.000	0.125	0.750
10.000	0.125	1.000

Formula Used

For a continuous uniform random variable on the interval [a, b], every value inside the range is equally likely.

Probability density function: f(x) = 1 / (b − a), for a ≤ x ≤ b, otherwise 0.

Cumulative distribution function: F(x) = 0 for x ≤ a, F(x) = (x − a)/(b − a) for a < x < b, and F(x) = 1 for x ≥ b.

Mean: μ = (a + b) / 2

Variance: σ² = (b − a)² / 12

Standard deviation: σ = √[(b − a)² / 12]

Interval probability: P(c ≤ X ≤ d) = F(d) − F(c)

Quantile: Q(p) = a + p(b − a)

How to Use This Calculator

Enter the lower bound a and upper bound b.
Provide a point x for PDF and CDF evaluation.
Enter x1 and x2 to measure interval probability.
Type a probability p between 0 and 1.
Set graph points for smoother or lighter plotting.
Click Calculate to display results above the form.
Review the graph, summary metrics, and example table.
Use CSV or PDF buttons to export results.

Frequently Asked Questions

1. What does a continuous uniform distribution represent?

It represents a variable that can take any value between two limits, where each equal-length interval has the same probability.

2. Why is the PDF constant inside the interval?

Because the distribution assumes equal likelihood across the entire range, the density stays flat from the lower bound to the upper bound.

3. Can x be outside the interval [a, b]?

Yes. The PDF becomes zero outside the interval. The CDF becomes zero below a and one above b.

4. How is interval probability calculated?

It is calculated by subtracting the cumulative probability at the lower interval value from the cumulative probability at the upper interval value.

5. What does the quantile output mean?

The quantile returns the x-value where the chosen cumulative probability is reached within the distribution range.

6. Why must b be greater than a?

The interval width must be positive. Otherwise, the density formula divides by zero or produces an invalid distribution.

7. What are common applications of this distribution?

It is used in simulation, random sampling, modeling uncertain measurements, timing problems, and simple probability examples.

8. Do single points have probability in continuous models?

No. A single exact point has probability zero, even though the density at that point can still be positive.