Use our Triangular Distribution Calculators to calculate the PDF, CDF, Mean, Mode, Median, Variance and generate sample

Triangular Distribution Calculator

Triangular Distribution Calculator

For parameters a (lower bound), b (upper bound) and c (mode with a<c<b), the PDF is given by:

f(x)={0,x<a or x>b,2(x−a)(b−a)(c−a),a≤x≤c,2(b−x)(b−a)(b−c),c≤x≤b.

The CDF is:

F(x)={0,x<a,(x−a)2(b−a)(c−a),a≤x≤c,1−(b−x)2(b−a)(b−c),c<x≤b,1,x>b.

Step 1: Enter Parameters

Enter the minimum value (e.g., 0)

Enter the maximum value (e.g., 10)

Enter the mode (must satisfy a<c<b, e.g., 5)

Enter a value in [a, b] to evaluate the PDF and CDF

Triangular Distribution:
Expected Value: μ=a+b+c3
Variance: σ2=a2+b2+c2–ab–ac–bc18

Triangular Distribution Calculator (In-Depth Explanation)

Triangular Distribution Calculator (In-Depth Explanation)

The Triangular distribution is a continuous probability distribution defined by a lower limit a, an upper limit b, and a mode c (with a≤c≤b). It is often used as a simple model for uncertainty when limited sample data is available, and it features a piecewise linear probability density function that forms a triangle shape.

Table of Contents

  1. Overview of the Triangular Distribution
  2. Key Concepts
  3. Distribution Functions
  4. Step-by-Step Calculation Process
  5. Practical Examples
  6. Common Applications
  7. Conclusion

1. Overview of the Triangular Distribution

The Triangular distribution is defined by three parameters:

  • Minimum (a): The smallest possible value.
  • Mode (c): The most likely value, where the peak of the distribution occurs.
  • Maximum (b): The largest possible value.

The distribution is continuous on the interval [a,b] and is characterized by a linearly increasing density from a to c and a linearly decreasing density from c to b.

2. Key Concepts

Key points to understand when working with the Triangular distribution include:

  • Support: The distribution is defined only on the interval [a,b].
  • Shape: It is typically unimodal with a peak at c; if c equals the midpoint of [a,b], the distribution is symmetric.
  • Simplicity: The triangular distribution is useful when only the minimum, maximum, and most likely values are known.

3. Distribution Functions

Probability Density Function (PDF):

The PDF of the Triangular distribution is defined piecewise as follows:

f(x)={2(x−a)(b−a)(c−a),for a≤x≤c,2(b−x)(b−a)(b−c),for c<x≤b.

Cumulative Distribution Function (CDF):

The CDF is given by:

F(x)={(x−a)2(b−a)(c−a),for a≤x≤c,1−(b−x)2(b−a)(b−c),for c<x≤b.

The quantile function, while available in a piecewise form, is more complex and is typically computed numerically for a given probability p.

4. Step-by-Step Calculation Process

  1. Define the Parameters:

    Identify the minimum (a), mode (c), and maximum (b) values for your distribution.

  2. Determine the Interval:

    Decide whether your input value x lies in the interval [a,c] or (c,b] to select the correct formula.

  3. Calculate the PDF:

    Substitute x, a, c, and b into the corresponding piecewise formula for the PDF:

    f(x)={2(x−a)(b−a)(c−a),for a≤x≤c,2(b−x)(b−a)(b−c),for c<x≤b.
  4. Calculate the CDF:

    Similarly, substitute x into the appropriate piecewise formula for the CDF:

    F(x)={(x−a)2(b−a)(c−a),for a≤x≤c,1−(b−x)2(b−a)(b−c),for c<x≤b.
  5. Interpret the Results:

    The computed PDF value represents the relative likelihood of x occurring, while the CDF gives the cumulative probability up to x.

5. Practical Examples

Example: Calculating PDF and CDF

Scenario: Assume a triangular distribution with a minimum a=0, mode c=5, and maximum b=10.

For x=3: (Since 3 is between a and c)

f(3)=2(3−0)(10−0)(5−0)=650=0.12
F(3)=(3−0)2(10−0)(5−0)=950=0.18

For x=7: (Since 7 is between c and b)

f(7)=2(10−7)(10−0)(10−5)=650=0.12
F(7)=1−(10−7)2(10−0)(10−5)=1−950=0.82

These calculations illustrate the piecewise nature of the triangular distribution.

6. Common Applications

  • Project Management: Used in PERT analysis for estimating task durations.
  • Risk Analysis: Modeling uncertainties when only limited data is available.
  • Simulation: Serving as a simple model for random variables in Monte Carlo simulations.
  • Quality Control: Estimating process outcomes with known bounds and a most-likely value.

7. Conclusion

The Triangular Distribution Calculator provides a straightforward method for evaluating the properties of a triangular distribution defined by its minimum, mode, and maximum values. By applying the piecewise formulas for the probability density and cumulative distribution functions:

f(x)={2(x−a)(b−a)(c−a),a≤x≤c,2(b−x)(b−a)(b−c),c<x≤b,
F(x)={(x−a)2(b−a)(c−a),a≤x≤c,1−(b−x)2(b−a)(b−c),c<x≤b,

users can effectively model and analyze uncertain outcomes in various applications ranging from project management to simulation studies.

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