Discrete Uniform Distribution Calculator

Calculate probabilities, statistics, and visualize discrete uniform distributions. Essential tool for probability and statistics.

Discrete Uniform Distribution: A random variable X has a discrete uniform distribution if it takes each of the values \(x_1, x_2, \ldots, x_n\) with equal probability \(\frac{1}{n}\).

For integer values from a to b: \(X \sim U(a, b)\) where \(P(X = k) = \frac{1}{b - a + 1}\) for \(k = a, a+1, \ldots, b\)

The smallest integer value in the distribution
Please enter a valid integer.
The largest integer value in the distribution
Maximum must be greater than minimum.
Dice Roll (1-6)
Random Digit (0-9)
Numbers 1-10
Months (1-12)
Coin Toss (0=Tails,1=Heads)
Deck of Cards
20-sided Die
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Probability Calculator
Enter an integer value between a and b
Probability Mass Function (PMF)
Cumulative Distribution Function (CDF)

Understanding Discrete Uniform Distribution

The discrete uniform distribution is one of the simplest probability distributions. It describes an experiment where there are a finite number of equally likely outcomes. Each value in the specified range has the same probability of occurring.

Mathematical Definition:

A random variable X has a discrete uniform distribution on the integers \(a, a+1, \ldots, b\) if:

\(P(X = k) = \frac{1}{b - a + 1}\) for \(k = a, a+1, \ldots, b\)

We write \(X \sim U(a, b)\) to indicate that X follows a discrete uniform distribution from a to b.

Key Properties

1

Equally Likely Outcomes: All values in the range have exactly the same probability of occurring. This is the defining characteristic of uniform distributions.

2

Finite Support: The distribution is only defined for a finite set of integer values between a and b inclusive. The size of this set is \(n = b - a + 1\).

3

Symmetry: When a and b are symmetric around their mean, the distribution is symmetric. The mean is always the midpoint: \(\frac{a + b}{2}\).

Formulas

Property Formula Explanation
Probability Mass Function \(P(X = k) = \frac{1}{b - a + 1}\) Constant probability for all values
Cumulative Distribution Function \(F(k) = P(X \le k) = \frac{k - a + 1}{b - a + 1}\) For \(k = a, a+1, \ldots, b\)
Mean (Expected Value) \(E[X] = \frac{a + b}{2}\) Average of minimum and maximum
Variance \(Var(X) = \frac{(b - a + 1)^2 - 1}{12}\) Measure of spread
Standard Deviation \(\sigma = \sqrt{Var(X)}\) Square root of variance

Common Applications

  • Games of Chance: Dice rolls, card draws, roulette wheels
  • Random Sampling: Selecting random items from a finite population
  • Computer Science: Random number generation, hash functions
  • Quality Control: Random inspection of items from a production line
  • Cryptography: Generating random keys from a finite set

Calculator Features:

  • Calculate all key statistics: mean, variance, standard deviation
  • Compute exact probabilities: P(X = k), P(X ≤ k), P(X ≥ k), P(k₁ ≤ X ≤ k₂)
  • Visualize Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)
  • Generate detailed probability tables
  • Pre-loaded examples for common use cases

Frequently Asked Questions

Discrete uniform distributions are defined for a finite set of distinct values (like integers), while continuous uniform distributions are defined for all values in a continuous interval. Discrete distributions have probability mass functions, while continuous distributions have probability density functions.

A fair die roll follows a discrete uniform distribution from 1 to 6. Each face (1, 2, 3, 4, 5, 6) has an equal probability of 1/6. This is a classic example of a discrete uniform distribution.

The standard discrete uniform distribution is defined for integer values. However, the concept can be extended to any finite set of discrete values (not necessarily integers or equally spaced). In such cases, the probability mass function would still assign equal probability to each value in the set.

For a true discrete uniform distribution, a and b should be integers since the distribution is defined over integer values. If you enter non-integer values, the calculator will round them to the nearest integers to maintain mathematical correctness.

For a discrete uniform distribution from a to b, the probability that X falls between k₁ and k₂ (inclusive) is: P(k₁ ≤ X ≤ k₂) = (k₂ - k₁ + 1) / (b - a + 1). You can use the "P(k₁ ≤ X ≤ k₂)" option in the Probability Calculator tab to compute this automatically.