What is the difference between discrete and continuous uniform distributions?

The Discrete Uniform distribution applies to a finite set of distinct outcomes (e.g., the integers 1, 2, 3, 4, 5, 6). The Continuous Uniform distribution applies to an infinite number of outcomes within a continuous range (e.g., any real number between 0.0 and 1.0). The discrete version uses a Probability Mass Function (PMF), while the continuous version uses a Probability Density Function (PDF).

What is a real-world example of the discrete uniform distribution?

Rolling a fair six-sided die is the most common example. The possible outcomes are the integers from 1 to 6. The total number of outcomes is n=6, and the probability of rolling any specific number (e.g., a '4') is exactly 1/6.

How is the probability calculated for a range of values?

For a range [m, n], you simply count the number of successful outcomes in that range and divide by the total number of possible outcomes. For example, the probability of rolling a number between 2 and 4 (inclusive) on a fair die is P(2 ≤ X ≤ 4). The successful outcomes are {2, 3, 4}, which is 3 outcomes. The total outcomes are 6. So the probability is 3/6 = 0.5.

Discrete Uniform Distribution Calculator | statistical-calculators.site

How to Use This Calculator

Step 1: Enter Your Parameters

Lower Bound (a): The smallest integer value in your set of equally likely outcomes (e.g., 1 for a standard die)
Upper Bound (b): The largest integer value in your set of equally likely outcomes (e.g., 6 for a standard die)
Range Start (m): The beginning of the specific range you want to calculate probability for
Range End (n): The ending of the specific range you want to calculate probability for

Step 2: Click "Calculate"

The calculator will instantly show you:

P(m ≤ X ≤ n): The probability that X falls within your specified range, shown as both a decimal and fraction
P(X = k): The probability of any single outcome in the distribution (always 1/n where n is the total number of outcomes)
Statistical measures: mean (the midpoint of your range), variance, and standard deviation
Visual charts: PMF (bar chart) showing equal probability bars for each outcome, and CDF (step function) showing cumulative probabilities

Step 3: Export Your Results

Use the buttons below the results to:

Save as PNG: Download charts and summary as an image for reports or presentations
Save as PDF: Generate a complete report with all calculations and visualizations
More Calculators: Explore other probability distribution calculators on our site

💡 Quick Tip: The discrete uniform distribution is perfect for modeling fair games of chance like dice rolls, card draws, or lottery numbers. If your range [m, n] extends outside the bounds [a, b], the calculator will automatically clamp the values. If m > n, the values will be swapped automatically.

Understanding the Discrete Uniform Distribution

The Discrete Uniform Distribution is a simple probability distribution where a finite number of integer outcomes are equally likely to occur. It is characterized by its simplicity and the assumption that every possible outcome has the exact same probability.

Key Formulas

Let the possible integer outcomes be from $a$ to $b$, inclusive. The total number of outcomes is $n = b - a + 1$.

Probability Mass Function (PMF): The probability of any single outcome $k$.

$$P(X=k) = \frac{1}{n} = \frac{1}{b - a + 1}$$

Cumulative Distribution Function (CDF): The probability that a random variable $X$ is less than or equal to a value $k$.

$$F(k) = P(X \le k) = \frac{\lfloor k \rfloor - a + 1}{n}$$

Mean / Expected Value: The average of the minimum and maximum values.

$E[X] = \frac{a+b}{2}$

Variance: A measure of the spread of the distribution.

$Var(X) = \frac{n^2 - 1}{12} = \frac{(b-a+1)^2 - 1}{12}$

Examples with Step-by-Step Answers

Example 1: Rolling a Fair Die

Problem: You roll a single, fair six-sided die. What is the probability of rolling a number between 2 and 4 (inclusive)?

1. Identify the parameters:

The possible outcomes are {1, 2, 3, 4, 5, 6}. So, the lower bound $a = 1$ and upper bound $b = 6$.
The total number of outcomes is $n = 6 - 1 + 1 = 6$.
We want to find the probability for the range $m=2$ to $n=4$.

2. Calculate the probability of a single event: $P(X=k) = \frac{1}{6}$

3. Count the successful outcomes: The integers in the range [2, 4] are {2, 3, 4}. There are 3 successful outcomes.

4. Calculate the total probability:
$P(2 \le X \le 4) = P(X=2) + P(X=3) + P(X=4) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = 0.5$

Answer: There is a 50% chance of rolling a number between 2 and 4.

Example 2: Choosing a Random Number

Problem: A random integer is selected from the set {10, 11, ..., 20}. What is the mean and variance of this distribution?

1. Identify the parameters:

Lower bound $a = 10$ and upper bound $b = 20$.
Total outcomes $n = 20 - 10 + 1 = 11$.

2. Calculate the Mean:

$E[X] = \frac{a+b}{2} = \frac{10+20}{2} = 15$

3. Calculate the Variance:

$Var(X) = \frac{n^2 - 1}{12} = \frac{11^2 - 1}{12} = \frac{121 - 1}{12} = \frac{120}{12} = 10$

Answer: The mean of the distribution is 15, and the variance is 10.

Discrete Uniform Distribution Calculator

Results