36  Logistic Distribution

The Logistic distribution resembles the Normal in shape but has heavier tails. Its CDF — the sigmoid function — is one of the most important functions in statistics and machine learning, forming the mathematical foundation of logistic regression and neural network activations.

Formally, the random variate \(X\) defined for all of \(\mathbb{R}\), is said to have a Logistic Distribution (i.e. \(X \sim \text{Logistic}(\mu, s)\)) with location parameter \(\mu \in \mathbb{R}\) and scale parameter \(s > 0\). In R, the built-in functions are dlogis(x, location = mu, scale = s), plogis, qlogis, and rlogis.

36.1 Probability Density Function

\[ f(x) = \frac{e^{-(x-\mu)/s}}{s\,(1+e^{-(x-\mu)/s})^2} = \frac{F(x)\,[1-F(x)]}{s} \]

The figure below shows examples of the Logistic Probability Density Function for different parameter combinations.

Code 
par(mfrow = c(2, 2))
x <- seq(-8, 12, length = 500)

plot(x, dlogis(x, location = 0, scale = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 0, ",  ", s == 1)))

plot(x, dlogis(x, location = 0, scale = 2), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 0, ",  ", s == 2)))

plot(x, dlogis(x, location = 2, scale = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == 2, ",  ", s == 1)))

plot(x, dlogis(x, location = -1, scale = 0.5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(mu == -1, ",  ", s == 0.5)))

par(mfrow = c(1, 1))
Figure 36.1: Logistic Probability Density Function for various parameter combinations

36.2 Purpose

The Logistic distribution is a symmetric, unimodal distribution on \(\mathbb{R}\) that closely resembles the Normal but has heavier tails. Its CDF — the logistic sigmoid function — plays a fundamental role in binary classification models. Common applications include:

  • Logistic regression: the latent continuous variable whose sign determines the binary outcome
  • Neural networks: the sigmoid activation function is the Logistic CDF
  • Growth curve modeling (logistic growth) in biology and ecology
  • Bioassay and dose-response modeling where response probability increases with dose
  • Heavy-tailed alternative to the Normal for symmetric continuous data

Relation to the discrete setting. The Logistic distribution is the continuous analog of the Binomial distribution via the logit link: the Binomial models discrete 0/1 outcomes, while the Logistic models their continuous log-odds representation. In logistic regression, binary Binomial outcomes are modeled with a Logistic-distributed latent variable.

36.3 Distribution Function

\[ F(x) = \frac{1}{1+e^{-(x-\mu)/s}} \]

This is the logistic sigmoid function, one of the most widely used functions in statistics and machine learning.

The figure below shows the Logistic Distribution Function for \(\mu = 0\) and \(s = 1\).

Code 
x <- seq(-8, 8, length = 500)
plot(x, plogis(x, location = 0, scale = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Logistic Distribution Function",
     sub = expression(paste(mu == 0, ",  ", s == 1)))
Figure 36.2: Logistic Distribution Function (location = 0, scale = 1)

36.4 Moment Generating Function

\[ M_X(t) = e^{\mu t}\,\frac{\pi s t}{\sin(\pi s t)}, \quad |t| < \frac{1}{s} \]

36.5 1st Uncentered Moment

\[ \mu_1' = \mu \]

36.6 2nd Uncentered Moment

\[ \mu_2' = \mu^2 + \frac{s^2\pi^2}{3} \]

36.7 3rd Uncentered Moment

\[ \mu_3' = \mu^3 + \mu s^2\pi^2 \]

36.8 4th Uncentered Moment

\[ \mu_4' = \mu^4 + 2\mu^2 s^2\pi^2 + \frac{7s^4\pi^4}{15} \]

36.9 2nd Centered Moment

\[ \mu_2 = \frac{s^2\pi^2}{3} \]

36.10 3rd Centered Moment

\[ \mu_3 = 0 \]

The distribution is symmetric about \(\mu\), so all odd centered moments vanish.

36.11 4th Centered Moment

\[ \mu_4 = \frac{7s^4\pi^4}{15} \]

36.12 Expected Value

\[ \text{E}(X) = \mu \]

36.13 Variance

\[ \text{V}(X) = \frac{s^2\pi^2}{3} \]

36.14 Median

\[ \text{Med}(X) = \mu \]

36.15 Mode

\[ \text{Mo}(X) = \mu \]

36.16 Coefficient of Skewness

\[ g_1 = 0 \]

The distribution is symmetric about \(\mu\).

36.17 Coefficient of Kurtosis

\[ g_2 = \frac{\mu_4}{\mu_2^2} = \frac{7s^4\pi^4/15}{s^4\pi^4/9} = \frac{63}{15} = \frac{21}{5} = 4.2 \]

The excess kurtosis is \(g_2 - 3 = 1.2\), indicating heavier tails than the Normal distribution.

36.18 Parameter Estimation

Maximum likelihood estimates are obtained numerically in R:

library(MASS)

set.seed(42)
x_obs <- rlogis(100, location = 70, scale = 5)

fit <- fitdistr(x_obs, "logistic")
print(fit)
    location      scale   
  70.7404214    5.6983159 
 ( 0.9767560) ( 0.4816192)

36.19 R Module

36.19.1 RFC

The Logistic Distribution module is available in RFC under the menu “Distributions / Logistic Distribution”.

36.19.3 R Code

The following code demonstrates Logistic probability calculations:

mu <- 70; s <- 5

# Probability density function
dlogis(60, location = mu, scale = s)

# Distribution function: P(X <= 60)
plogis(60, location = mu, scale = s)

# Quantile function (median)
qlogis(0.5, location = mu, scale = s)

# Generate random Logistic numbers
set.seed(42)
rlogis(10, location = mu, scale = s)

# Standard deviation
cat("SD:", s * pi / sqrt(3), "\n")
[1] 0.02099872
[1] 0.1192029
[1] 70
 [1] 81.86891 83.50413 65.42896 77.94401 72.91474 70.38210 75.14155 60.69844
 [9] 73.24960 74.35767
SD: 9.068997

36.20 Example

Exam scores in a large class are modeled as \(X \sim \text{Logistic}(\mu = 70, s = 5)\). We compute the probability of scoring below 60.

mu <- 70; s <- 5

# P(score < 60)
cat("P(score < 60):", plogis(60, location = mu, scale = s), "\n")

# P(score > 80)
cat("P(score > 80):", 1 - plogis(80, location = mu, scale = s), "\n")

# Mean, median, SD
cat("Mean (= median):", mu, "\n")
cat("SD:", round(s * pi / sqrt(3), 4), "\n")
P(score < 60): 0.1192029 
P(score > 80): 0.1192029 
Mean (= median): 70 
SD: 9.069
Interactive Shiny app (click to load).
Open in new tab

36.21 Random Number Generator

Logistic random variates are generated via the inverse-CDF method. Since \(F(x) = 1/(1+e^{-(x-\mu)/s})\), the quantile function is the log-odds (logit) function:

\[ X = \mu - s\ln\!\left(\frac{1}{U} - 1\right) \sim \text{Logistic}(\mu, s) \quad \text{when } U \sim \text{U}(0,1) \]

set.seed(123)
n <- 1000; mu <- 0; s <- 1

# Inverse-transform method
u <- runif(n)
x_inv <- mu - s * log(1/u - 1)

# Compare with rlogis
x_rlogis <- rlogis(n, location = mu, scale = s)

cat("Inverse-transform: mean =", round(mean(x_inv), 4),
    "  var =", round(var(x_inv), 4), "\n")
cat("rlogis():          mean =", round(mean(x_rlogis), 4),
    "  var =", round(var(x_rlogis), 4), "\n")
cat("Theoretical:       mean =", mu,
    "  var =", round(s^2 * pi^2 / 3, 4), "\n")
Inverse-transform: mean = -0.0202   var = 3.2444 
rlogis():          mean = -0.0087   var = 3.2559 
Theoretical:       mean = 0   var = 3.2899
Interactive Shiny app (click to load).
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36.22 Property 1: Sigmoid CDF

The CDF \(F(x) = 1/(1+e^{-(x-\mu)/s})\) is the logistic sigmoid function — the foundation of logistic regression. The log-odds (logit) of \(F(x)\) is linear in \(x\):

\[ \text{logit}(F(x)) = \ln\!\frac{F(x)}{1-F(x)} = \frac{x - \mu}{s} \]

36.23 Property 2: Self-Referential Density

The density can be expressed in terms of the CDF:

\[ f(x) = \frac{F(x)\,[1-F(x)]}{s} \]

This elegant identity is unique to the Logistic distribution and is exploited in the derivation of the logistic regression score equations.

36.24 Property 3: Heavier Tails than Normal

The Logistic distribution has kurtosis \(g_2 = 4.2\) versus \(g_2 = 3\) for the Normal. This means the Logistic assigns more probability to values far from the mean — a feature that makes it more robust to outliers in some applications.