Uniform Distribution Calculator
Calculate probabilities, PDF, CDF, and statistical measures for uniform distribution
Distribution Parameters
Minimum value of the distribution
Maximum value of the distribution (must be > a)
Results
Distribution: U(0, 1)
PDF: f(x) = 1/(b-a) = 1.000000 for 0 ≤ x ≤ 1
CDF: F(x) = (x-a)/(b-a) for 0 ≤ x ≤ 1
Mean: μ = (a+b)/2 = 0.500
Variance: σ² = (b-a)²/12 = 0.083333
Example Calculation
Standard Uniform Distribution
Distribution: U(0, 1)
Parameters: a = 0, b = 1
PDF: f(x) = 1 for 0 ≤ x ≤ 1
CDF: F(x) = x for 0 ≤ x ≤ 1
Probability Examples
P(X ≤ 0.5) = 0.5/1 = 0.5 (50%)
P(0.2 ≤ X ≤ 0.8) = (0.8-0.2)/(1-0) = 0.6 (60%)
P(X > 0.7) = 1 - 0.7 = 0.3 (30%)
Distribution Properties
Rectangular Shape
Constant probability density
Symmetric
Mean equals median
Bounded
Defined on finite interval
Key Formulas
f(x) = 1/(b-a) for a ≤ x ≤ b
F(x) = (x-a)/(b-a) for a ≤ x ≤ b
μ = (a+b)/2
σ² = (b-a)²/12
Q(p) = (b-a)×p + a
Understanding the Uniform Distribution
What is the Uniform Distribution?
The uniform distribution is a continuous probability distribution where all outcomes in a given interval [a,b] are equally likely. It's also known as the rectangular distribution because its probability density function forms a rectangle.
Key Characteristics
- •Constant probability density over the interval
- •Zero probability density outside the interval
- •Symmetric around the midpoint (a+b)/2
- •Skewness is always zero
Applications
- •Random number generation
- •Monte Carlo simulations
- •Modeling equally likely outcomes
- •Quality control processes
- •Statistical hypothesis testing
Note: The standard uniform distribution U(0,1) is the foundation for generating random numbers from other distributions using transformation methods.
Distribution Types
Standard Uniform
U(0,1) - Most common form used in random number generation
General Uniform
U(a,b) - Defined on any interval [a,b] where a < b
Discrete Uniform
Equal probability for finite discrete outcomes