The triangular distribution is a continuous probability distribution with a probability density function shaped like a triangle.
It is defined by three values:
- The minimum value a
- The maximum value b
- The peak value c
The name of the distribution comes from the fact that the probability density function is shaped like a triangle.
It turns out that this distribution is extremely useful in the real world because we can often estimate the minimum value (a), the maximum value (b), and the most likely value (c) that a random variable will take on, so we can often model the behavior of random variables by using a triangular distribution with the knowledge of just these three values.
For example, a restaurant might estimate that their total sales for the upcoming week will be a minimum of $10,000, a maximum of $30,000, and most likely $25,000.
Using just these three numbers, they could use a triangular distribution to find the probability that they’ll achieve a certain number of sales.
Properties of the Triangular Distribution
The triangular distribution has the following properties:
PDF:
CDF:
Mean: (a + b + c) / 3
Mode: c
Variance: (a2 + b2 + c2 – ab – ac – bc) / 18
Example of Using the Triangular Distribution
Let’s return to the example from earlier. Suppose a restaurant estimates that their total sales for the upcoming week will be a minimum of $10,000, a maximum of $30,000, and most likely $25,000.
What is the probability that the restaurant makes less than $20,000 total sales?
To answer this, we can let x = total sales. We know that x is between the minimum value a of $10k and the peak value c of $25k.
Thus, according to the PDF we can use the following equation to find the probability that the restaurant makes less than $20,000 total sales:
- P(X < $20,000) = (x-a)2 / ((b-a)(c-a))
- P(X < $20,000) = (20,000-10,000)2 / ((30,000-10,000)(25,000-10,000))
- P(X < $20,000) = .333
The probability that the restaurant makes less than $20,000 total sales is .333.
What is the mean expected sales for the restaurant?
We can calculate the mean expected sales using the formula for the mean given earlier:
- Mean = (a + b + c) / 3
- Mean = ($10,000 + $30,000 + $25,000) / 3
- Mean = $21,667
The mean expected sales is 21,667.
Additional Resources
The following tutorials provide introductions to other commonly used distributions:
The Normal Distribution
The Binomial Distribution
The Poisson Distribution
The Geometric Distribution
Where does the triangular distribution formula come from?
Hi Barry…The triangular distribution is a probability distribution that is defined by three parameters: the minimum value (a), the maximum value (b), and the mode (c). It is often used in situations where the distribution of data is unknown but a rough estimate of the minimum, maximum, and most likely values is available. The triangular distribution is particularly useful in simulation and project management (e.g., PERT analysis).
### Derivation of the Triangular Distribution Formula
The triangular distribution is essentially a piecewise linear distribution, which means that its probability density function (PDF) is defined by two linear functions over its domain. Here’s how the formula comes about:
1. **Basic Parameters**:
– \( a \): The minimum value.
– \( b \): The maximum value.
– \( c \): The mode or the most likely value, where \( a \leq c \leq b \).
2. **Piecewise Definition**:
The PDF of the triangular distribution is defined in two parts:
– For \( x \) between \( a \) and \( c \), the distribution increases linearly.
– For \( x \) between \( c \) and \( b \), the distribution decreases linearly.
3. **Linear Functions**:
The increasing and decreasing linear functions are defined to ensure that the total area under the curve (which represents the probability) is equal to 1.
– **For \( x \) in [a, c]**:
\[
f(x) = \frac{2(x – a)}{(b – a)(c – a)}
\]
This is derived by noting that the slope of the line from \( (a, 0) \) to \( (c, \frac{2}{b-a}) \) must result in an increasing probability.
– **For \( x \) in [c, b]**:
\[
f(x) = \frac{2(b – x)}{(b – a)(b – c)}
\]
This is the decreasing side of the triangle, derived by ensuring that the slope from \( (c, \frac{2}{b-a}) \) to \( (b, 0) \) matches the probability density.
### **Complete PDF**:
\[
f(x) =
\begin{cases}
\frac{2(x – a)}{(b – a)(c – a)} & \text{for } a \leq x < c \\ \frac{2(b - x)}{(b - a)(b - c)} & \text{for } c \leq x \leq b \\ 0 & \text{otherwise} \end{cases} \] ### **Where It Comes From**: The triangular distribution can be seen as a simple model to approximate more complex distributions, especially when data is sparse or only limited information is available. It's called "triangular" because of the shape of its PDF, which forms a triangle. The formula comes from the need to: - Have a distribution that is bounded by \( a \) and \( b \). - Ensure that the mode \( c \) represents the most likely value. - Ensure that the total probability is 1 (i.e., the area under the curve is 1). This distribution is often used in decision analysis and project management where the exact distribution of an outcome is unknown, but a range of values with a most likely estimate is available.