Uniform Distribution Calculator
Calculate probability density, cumulative probability, and properties of the continuous uniform distribution on [a, b].
PDF f(x)
0.10000000
PDF f(x) vs Lower Bound (a)
Formule
## How to Calculate Uniform Distribution Properties ### Formulas **PDF: f(x) = 1 / (b - a)** for a <= x <= b **CDF: P(X <= x) = (x - a) / (b - a)** The continuous uniform distribution assigns equal probability density to all values in the interval [a, b]. The probability of falling in any sub-interval depends only on the sub-interval's length relative to the total range. The mean is the midpoint (a+b)/2.
Exemple Résolu
A bus arrives uniformly between 0 and 10 minutes. What is P(arrival <= 7 min)?
- 01PDF = 1 / (10 - 0) = 0.1
- 02CDF = (7 - 0) / (10 - 0) = 0.7
- 03P(X <= 7) = 70%
- 04Mean = (0 + 10) / 2 = 5
- 05Variance = (10 - 0)^2 / 12 = 100/12 ≈ 8.3333
Questions Fréquentes
Why is the PDF constant?
Because every value in [a,b] is equally likely. The height 1/(b-a) ensures the total area under the curve equals 1, as required for a valid probability distribution.
How do I find P(c < X < d) for a uniform distribution?
P(c < X < d) = (d - c) / (b - a), as long as a <= c < d <= b. The probability depends only on the length of the interval relative to the total range.
What is the discrete uniform distribution?
The discrete version assigns equal probability 1/n to each of n possible outcomes. For example, a fair die has a discrete uniform distribution on {1, 2, 3, 4, 5, 6} with P(x) = 1/6.
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