Enter the minimum value (a), maximum value (b), and a specific value of x to compute probabilities for a Discrete Uniform Distribution. You get back the point probability P(X = x), all five cumulative probabilities, plus the mean, variance, and standard deviation of the distribution.
P(X = x)
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P(X ≤ x)
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P(X < x)
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P(X ≥ x)
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P(X > x)
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Mean E(X)
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Variance V(X)
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Standard Deviation
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A discrete uniform distribution is a probability distribution where every integer value between a minimum (a) and maximum (b), inclusive, is equally likely. A classic example is rolling a fair six-sided die, where each face (1 through 6) has exactly a 1/6 probability of appearing.
The probability of any single value x in the range [a, b] is P(X = x) = 1 / (b − a + 1). Every value in the range shares the same probability, so you simply divide 1 by the total number of possible outcomes.
The cumulative probability P(X ≤ x) = (x − a + 1) / (b − a + 1) counts how many values from a up to x are possible and divides by the total count. From this you can derive P(X < x) = (x − a) / (b − a + 1), P(X ≥ x) = (b − x + 1) / (b − a + 1), and P(X > x) = (b − x) / (b − a + 1).
The mean of a discrete uniform distribution is E(X) = (a + b) / 2. It is simply the midpoint of the interval [a, b]. For example, rolling a fair die with a = 1 and b = 6 gives a mean of (1 + 6) / 2 = 3.5.
The variance is V(X) = ((b − a + 1)² − 1) / 12. For a standard die (a = 1, b = 6), this gives V(X) = (36 − 1) / 12 = 35/12 ≈ 2.9167. The standard deviation is the square root of the variance.
The value x must be an integer that falls within the range [a, b]. If x is outside that range, P(X = x) = 0 and the cumulative probabilities are either 0 or 1 depending on whether x is below the minimum or above the maximum.
In a discrete uniform distribution, the variable takes on a finite set of equally spaced integer values (e.g., 1, 2, 3, 4, 5, 6). In a continuous uniform distribution, the variable can take any real value within a continuous interval [a, b]. For the discrete version, point probabilities are non-zero; for the continuous version, P(X = x) = 0 for any single point.
Common examples include rolling a fair die (values 1–6), drawing a random card from a shuffled deck of numbered cards, randomly selecting a day of the week, or generating a random integer in a computer simulation. Any situation where all outcomes are integers and equally likely follows a discrete uniform distribution.