Understanding the Triangular Distribution and Its Application in R

rtip

distribution

Author

Steven P. Sanderson II, MPH

Published

November 1, 2023

Introduction

As an R programmer and enthusiast, I’m excited to delve into the fascinating world of probability distributions. One of the lesser-known but incredibly useful distributions is the Triangular Distribution, and today we’ll explore what it is and how to leverage it in R using the EnvStats library.

What is the Triangular Distribution?

The Triangular Distribution is a continuous probability distribution with a triangular shape, hence the name. It is defined by three parameters: min, max, and mode. These parameters determine the range of values the distribution can take and the most likely value within that range. In mathematical terms, the probability density function (PDF) of the Triangular Distribution is given by:

f(x) = (2 / (b - a)) * (x - a) / (c - a) for a ≤ x < c
f(x) = (2 / (b - a)) * (b - x) / (b - c) for c ≤ x ≤ b

Where: - a is the minimum value (min parameter). - b is the maximum value (max parameter). - c is the mode, which is the peak or most likely value (mode parameter).

Using the EnvStats R Library

To work with the Triangular Distribution in R, we can use the functions provided by the EnvStats library. Here are the key functions you need to know:

dtri(x, min = 0, max = 1, mode = 1/2): This function calculates the probability density at a given x. You can specify the min, max, and mode parameters to define the distribution.

ptri(q, min = 0, max = 1, mode = 1/2): Use this function to find the cumulative probability up to a given q. Again, you can customize the min, max, and mode parameters.

qtri(p, min = 0, max = 1, mode = 1/2): The quantile function, which helps you find the value of x for a given cumulative probability p. As always, you can set min, max, and mode to match your specific distribution.

rtri(n, min = 0, max = 1, mode = 1/2): This function generates a random set of n numbers following the Triangular Distribution with the specified parameters.

Practical Example in R

Let’s see how to use these functions in a practical example. Suppose we want to model the distribution of daily temperatures in a specific region. We have historical data indicating that the minimum temperature is -5°C, the maximum temperature is 30°C, and the most likely temperature (mode) is around 20°C.

Here’s how you can work with this scenario in R using the EnvStats library:

# Load the EnvStats librarylibrary(EnvStats)# Define the parametersmin_temp <--5max_temp <-30mode_temp <-20# Calculate the density at x = 15°Cdensity_at_15 <-dtri(15, min = min_temp, max = max_temp, mode = mode_temp)cat("Density at 15°C:", density_at_15, "\n")

Density at 15°C: 0.04571429

# Calculate the cumulative probability up to 25°Ccumulative_prob_up_to_25 <-ptri(25, min = min_temp, max = max_temp, mode = mode_temp)cat("Cumulative Probability up to 25°C:", cumulative_prob_up_to_25, "\n")

Cumulative Probability up to 25°C: 0.9285714

# Find the temperature value for a cumulative probability of 0.75temperature_for_prob_0.75<-qtri(0.75, min = min_temp, max = max_temp, mode = mode_temp)cat("Temperature for Cumulative Probability 0.75:", temperature_for_prob_0.75, "\n")

Temperature for Cumulative Probability 0.75: 20.64586

# Generate a random set of 10 temperaturesrandom_temperatures <-rtri(10, min = min_temp, max = max_temp, mode = mode_temp)cat("Random Temperatures:", random_temperatures, "\n")

In this example, we’ve used the Triangular Distribution to model daily temperatures, calculate probabilities, find quantiles, and generate random temperature values.

The Triangular Distribution is a versatile tool for modeling scenarios where you have some knowledge about the range and likelihood of an event or outcome. Whether you’re simulating real-world scenarios or conducting risk assessments, the EnvStats library in R makes it easy to work with this distribution.

So, the next time you need to model uncertain events with known bounds and modes, remember the Triangular Distribution and its helpful functions in R!