Chapter 3 – Simple Distributions

Monte Carlo simulations rely on probability distributions to model uncertainty.

💡 What do we mean by uncertainty?
In GMP and pharmaceutical work, uncertainty means we cannot predict exactly what value will occur —
only the range and likelihood of possible values.

Examples:

• The assay of an API: usually near 98%, but with natural lab variability.
• The weight of a tablet: centered at 100 mg, but not every tablet weighs exactly 100 mg.
• The time to equipment failure: unpredictable, but often follows a statistical pattern.

Probability distributions are the tools we use to describe and simulate this uncertainty.

In this chapter, we review four common distributions used in GMP & Pharma applications, with visual examples.

📊 1. Uniform Distribution

• Definition: All values in a given range have the same probability.
• Parameters: a = minimum, b = maximum.
• R Example:

  set.seed(123)
  runif(50, min = 40, max = 60)  # 50 values between 40 and 60
  

• Use in Pharma: Modeling tolerance bands where all outcomes within limits are equally likely.

Figure 3.1 – Uniform distribution (example with n=50).

📊 2. Normal Distribution

• Definition: The famous bell curve, symmetric around the mean.

• Parameters: μ = mean, σ = standard deviation.

• R Example:

  set.seed(123)
  rnorm(50, mean = 0, sd = 1)  # 50 values from the standard Normal distribution (mean=0, sd=1)
  

• Use in Pharma: Measurement errors, assay results, tablet weights.

Figure 3.2 – Normal distribution (example with n=50).

📊 3. Exponential Distribution

• Definition: Models the time between independent events.

• Parameter: λ = rate (events per time unit).

• R Example:

  set.seed(123)
  rexp(50, rate = 1)  # 50 exponential values with λ = 1
  

• Use in Pharma: Time to microbial contamination, equipment failure.

Figure 3.3 – Exponential distribution (example with n=50).

📊 4. Triangular Distribution

• Definition: Defined by a minimum (a), a maximum (b), and a most likely value (c).

• R Example (using triangle package):

  library(triangle)  # make sure the package is installed
  set.seed(123)
  rtriangle(50, a = 0, b = 3, c = 1)
  

• Use in Pharma: When only expert estimates are available, specified as minimum (a), maximum (b), and most likely (c).

Figure 3.4 – Triangular distribution (example with n=50).

📌 Choosing a Distribution – Quick Guide

🔎 Note: In the small-sample examples (n = 50), distributions may look irregular due to random variation.
With larger samples (see illustrative examples, n = 1000), the histograms converge toward the smooth theoretical shapes.
This reflects a key principle of Monte Carlo simulation: results depend on the number of random draws.

📊 Illustrative Examples

Instead of looking at just 50 numbers, we can simulate 1,000 values and visualize them.

Normal Distribution

set.seed(123)
x <- rnorm(1000, mean = 100, sd = 5)

hist(x,
     main = "Normal Distribution (mean = 100, sd = 5)",
     xlab = "Value",
     col = "lightblue",
     border = "white")

The histogram shows the classic bell-shaped curve, centered at 100 with most values within ±10.

Figure 3.5 – Histogram of 1,000 simulated values from a Normal distribution (mean=100, sd=5)

Triangular Distribution

library(triangle)
set.seed(123)
y <- rtriangle(1000, a = 95, b = 105, c = 100)

hist(y,
     main = "Triangular Distribution (a=95, b=105, c=100)",
     xlab = "Value",
     col = "lightgreen",
     border = "white")

The histogram shows a triangle shape, rising towards the most likely value (100) and falling off towards the extremes.

Figure 3.6 – Histogram of 1,000 simulated values from a Triangular distribution (a=95, b=105, c=100)

💊 Why Distributions Matter in GMP

Choosing the right distribution ensures that simulated data reflect real-world processes.
Using the wrong one can lead to misleading conclusions, especially in risk assessment and capability analysis.

👉 In the next chapter, we will see how distributions combine through the transfer equation to model variability propagation in real processes.