Beta Distribution Calculator
Calculate probabilities, PDF, CDF, and statistical measures for Beta distribution
Beta Distribution Parameters
Must be positive. Controls left side of distribution.
Must be positive. Controls right side of distribution.
Distribution Shape
Beta(2, 5) - Right-skewed
→ Skewed distribution (α < β)
Calculation Mode
Value must be between 0 and 1 for Beta distribution
Results
Calculation Details
Distribution: Beta(2, 5)
Shape: Right-skewed
Mean: 0.2857
Std Dev: 0.1597
Beta Distribution Properties
Key Features
- • Support: [0, 1] interval
- • Two shape parameters: α, β > 0
- • Very flexible distribution family
- • Can model many different shapes
Current Distribution
Alpha (α): 2
Beta (β): 5
Shape: Right-skewed
Mean: 0.2857
Common Examples
Understanding the Beta Distribution
What is Beta Distribution?
The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It's characterized by two positive shape parameters α (alpha) and β (beta) that control the distribution's shape. The Beta distribution is extremely flexible and can take many different shapes including uniform, U-shaped, bell-shaped, and J-shaped distributions.
Key Properties
- •Support: [0, 1] - values between 0 and 1
- •Parameters: α > 0, β > 0 (shape parameters)
- •Symmetric when α = β
- •Right-skewed when α < β
- •Left-skewed when α > β
Mathematical Foundation
Probability Density Function
f(x) = x^(α-1) × (1-x)^(β-1) / B(α,β)
where B(α,β) = Γ(α)Γ(β) / Γ(α+β) is the Beta function
Applications
- •Bayesian statistics (conjugate prior)
- •Modeling proportions and percentages
- •Project management (PERT)
- •Quality control and reliability
Distribution Shapes
Symmetric Shapes (α = β):
- • α = β = 1: Uniform distribution
- • α = β = 0.5: U-shaped
- • α = β = 2: Parabolic
- • α = β > 2: Bell-shaped
Asymmetric Shapes:
- • α < β: Right-skewed
- • α > β: Left-skewed
- • α = 1, β > 1: Decreasing
- • α > 1, β = 1: Increasing