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:

$$ f(x) = \begin{cases} 0, & x < a \text{ or } x > b, \\\\ \dfrac{2(x-a)}{(b-a)(c-a)}, & a \le x \le c, \\\\ \dfrac{2(b-x)}{(b-a)(b-c)}, & c \le x \le b. \end{cases} $$

The CDF is:

$$ F(x) = \begin{cases} 0, & x < a, \\\\ \dfrac{(x-a)^2}{(b-a)(c-a)}, & a \le x \le c, \\\\ 1-\dfrac{(b-x)^2}{(b-a)(b-c)}, & c < x \le b, \\\\ 1, & x > b. \end{cases} $$

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: $$ \mu = \frac{a+b+c}{3} $$
Variance: $$ \sigma^2 = \frac{a^2+b^2+c^2 – ab – ac – bc}{18} $$

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 \( a \leq c \leq b \)). 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:

\( f(x) = \begin{cases} \dfrac{2(x-a)}{(b-a)(c-a)}, & \text{for } a \leq x \leq c, \\ \\ \dfrac{2(b-x)}{(b-a)(b-c)}, & \text{for } c < x \leq b. \end{cases} \)

Cumulative Distribution Function (CDF):

The CDF is given by:

\( F(x) = \begin{cases} \dfrac{(x-a)^2}{(b-a)(c-a)}, & \text{for } a \leq x \leq c, \\ \\ 1 - \dfrac{(b-x)^2}{(b-a)(b-c)}, & \text{for } c < x \leq b. \end{cases} \)

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:

    \( f(x) = \begin{cases} \dfrac{2(x-a)}{(b-a)(c-a)}, & \text{for } a \leq x \leq c, \\ \\ \dfrac{2(b-x)}{(b-a)(b-c)}, & \text{for } c < x \leq b. \end{cases} \)
  4. Calculate the CDF:

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

    \( F(x) = \begin{cases} \dfrac{(x-a)^2}{(b-a)(c-a)}, & \text{for } a \leq x \leq c, \\ \\ 1 - \dfrac{(b-x)^2}{(b-a)(b-c)}, & \text{for } c < x \leq b. \end{cases} \)
  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 \( c \))

\( f(3) = \frac{2(3-0)}{(10-0)(5-0)} = \frac{6}{50} = 0.12 \)
\( F(3) = \frac{(3-0)^2}{(10-0)(5-0)} = \frac{9}{50} = 0.18 \)

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

\( f(7) = \frac{2(10-7)}{(10-0)(10-5)} = \frac{6}{50} = 0.12 \)
\( F(7) = 1 - \frac{(10-7)^2}{(10-0)(10-5)} = 1 - \frac{9}{50} = 0.82 \)

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:

\( f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)}, & a \leq x \leq c, \\ \frac{2(b-x)}{(b-a)(b-c)}, & c < x \leq b, \end{cases} \)
\( F(x) = \begin{cases} \frac{(x-a)^2}{(b-a)(c-a)}, & a \leq x \leq c, \\ 1 - \frac{(b-x)^2}{(b-a)(b-c)}, & c < x \leq b, \end{cases} \)

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

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