SMp(x) Distribution Calculator
Simulate various probability distributions using the versatile SMp(x) function
Model Parameters
Minimum value of the variable x
Maximum value of the variable x
Value where function reaches maximum
Left side power parameter (must be > 0)
Right side power parameter (must be > 0)
Maximum value of the SMp(x) function
Calculate SMp(x) for specific x
Normal Distribution Detected
Your parameters approximate a normal distribution. Probability calculations are now available.
Calculating Probabilities
SMp(x) Formula
For x < PXmin: SMp(x) = 0
For PXmin ≤ x ≤ ML: SMp(x) = [(x - PXmin)/(ML - PXmin)]^p₁ × Max
For ML ≤ x ≤ Xmax: SMp(x) = [(Xmax - x)/(Xmax - ML)]^p₂ × Max
For x > Xmax: SMp(x) = 0
Distribution Sample Points
| x | SMp(x) |
|---|---|
| -3.000 | 0.0000 |
| -2.400 | 0.0160 |
| -1.800 | 0.0640 |
| -1.200 | 0.1440 |
| -0.600 | 0.2560 |
| 0.000 | 0.4000 |
| 0.600 | 0.2560 |
| 1.200 | 0.1440 |
| 1.800 | 0.0640 |
| 2.400 | 0.0160 |
| 3.000 | 0.0000 |
Parameter Guidelines
Valid Range
Ensure PXmin < ML < Xmax
Power Parameters
Both p₁ and p₂ must be > 0
Normal Distribution
Set p₁ = p₂ > 1 and ML = (PXmin + Xmax)/2
Common Applications
Normal distribution simulation
Poisson distribution modeling
Binomial distribution approximation
Exponential distribution modeling
Custom distribution design
Understanding the SMp(x) Distribution
What is SMp(x)?
The SMp(x) distribution is a versatile probability function that can simulate virtually any probability distribution using six parameters. It's particularly useful for modeling complex distributions that don't fit standard forms.
Key Parameters
- •PXmin: Lower boundary of the distribution
- •Xmax: Upper boundary of the distribution
- •ML: Mode - point of maximum probability
- •p₁, p₂: Shape parameters controlling distribution curve
- •Max: Maximum value of the function
Normal Distribution Conditions
For normal distribution approximation:
- • p₁ = p₂ > 1 (equal powers)
- • ML = (PXmin + Xmax)/2 (symmetric)
- • Max = (p₁ + 1) / [2(ML - PXmin)] (normalized)
Probability Calculations
Note: Probability calculations are only available when the SMp(x) parameters approximate a normal distribution.
Applications: Weather forecasting, quality control, risk assessment, and custom probability modeling.