Continuous Probability Tool

Uniform Distribution Calculator

Calculate probabilities and descriptive statistics for a continuous uniform distribution. Enter the minimum and maximum values of the distribution plus a point or interval to find PDF, CDF, interval probability, mean, variance, and standard deviation instantly.

PDF & CDF Interval Probability Mean & Variance Instant Results

Enter Distribution Values

Define the interval and probability range.

Minimum Value (a) Lower endpoint of the uniform distribution.

Maximum Value (b) Upper endpoint of the uniform distribution.

Point Probability

Enter x to calculate PDF and cumulative probabilities.

X Value (x) Point used for PDF and CDF calculations.

Support Valid range of the random variable.

Interval Probability

Calculate the probability that X falls between two values.

Range Start (c) Beginning of the probability interval.

Range End (d) End of the probability interval.

For a continuous uniform distribution, every interval of equal length inside [a,b] has the same probability. The probability of any single exact point is zero, even though the PDF has a positive density inside the interval.

Probability Guide

What Is a Uniform Distribution?

A continuous uniform distribution is a probability distribution in which all intervals of equal length within a specified range have equal probability. The distribution is defined by a lower endpoint a and an upper endpoint b.

The probability density remains constant between a and b and is zero outside that interval. This produces the familiar rectangular shape of the continuous uniform probability density function.

Why Is It Called Uniform?

It is called uniform because probability density is spread evenly throughout the interval. No region of the same width inside the valid range has more probability than another.

How It Works

How to Use the Uniform Distribution Calculator

Enter a and b

Enter the minimum value a and maximum value b that define the support of the distribution.

Enter x or a Range

Enter a point x for PDF and CDF calculations or provide c and d to calculate an interval probability.

View Results

The calculator returns probability density, cumulative probability, interval probability, mean, variance, and standard deviation.

Example: For a uniform distribution between 0 and 10, the probability that X lies between 2 and 7 is (7 − 2) / (10 − 0) = 0.5, or 50%.

Probability Functions

Uniform Distribution PDF and CDF

Probability Density Function

For a continuous uniform distribution from a to b, the probability density function is: f(x) = 1 / (b − a) for values of x between a and b. Outside the interval, the density is zero.

Cumulative Distribution Function

The cumulative distribution function measures the probability that X is less than or equal to a particular value x. Inside the support interval it is: F(x) = (x − a) / (b − a) .

For continuous distributions, the probability of one exact value is always zero. The PDF represents probability density, not the probability of a single point.

Distribution Statistics

Mean, Variance and Standard Deviation

The mean of a continuous uniform distribution is the midpoint of the interval: μ = (a + b) / 2 .

The variance measures how widely values are spread around the mean and is calculated as: σ² = (b − a)² / 12 . The standard deviation is the square root of the variance.

Interval Probability

How to Calculate Probability Between Two Values

For an interval that lies completely inside the distribution support, the probability between c and d is calculated by dividing the length of the desired interval by the total length of the distribution: P(c ≤ X ≤ d) = (d − c) / (b − a) .

If part of the requested interval lies outside the support, this calculator automatically clips it to the valid [a,b] range before calculating probability.

Comparison

Continuous vs Discrete Uniform Distribution

Feature	Continuous Uniform	Discrete Uniform
Possible Values	Any real value in an interval	A finite set of separate values
Single-Point Probability	0	Can be greater than 0
Probability Tool	Probability density	Probability mass
Typical Example	Random time within an interval	Fair die outcomes

Common Questions

Uniform Distribution Calculator FAQs

A uniform distribution means probability density is evenly distributed across the specified interval. Equal-width intervals therefore have equal probabilities.

The value a is the minimum or lower endpoint and b is the maximum or upper endpoint of the continuous uniform distribution.

Inside the interval from a to b, the probability density is constant and equals 1 divided by b minus a. Outside the interval, the density is zero.

The CDF gives the probability that the random variable is less than or equal to a specified value x.

A continuous random variable has infinitely many possible values. Probability is assigned to intervals rather than single points, so the probability of exactly one particular value is zero.

The mean is the midpoint between the minimum and maximum values and is calculated using (a + b) / 2.

Yes. The endpoints of a continuous uniform distribution may be negative, positive, or zero, as long as the maximum value b is greater than the minimum value a.

The calculations on this page run directly in your browser using JavaScript, so the entered values do not need to be sent to a calculation server.