Logistic Distribution probabilities using R
In this tutorial, you will learn about how to use dlogis()
, plogis()
, qlogis()
and rlogis()
functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Logistic distribution.
Before we discuss R functions for Logistic distribution, let us see what is Logistic distribution.
Logistic Distribution
Logistic distribution distribution is a continuous type probability distribution.
Let $X\sim Logis(\mu,\lambda)$. Then the probability density function of $X$ is
$$ \begin{aligned} f(x)&= \begin{cases} \frac{1}{\lambda}\frac{e^{\frac{(x-\mu)}{\lambda}}}{\bigg(1+e^{\frac{(x-\mu)}{\lambda}}\bigg)^2} , & -\infty < x< \infty; \\ & -\infty < \mu< \infty, \lambda>0;\\ 0, & Otherwise. \end{cases} \end{aligned} $$
where $\mu$ is the location parameter and $\lambda$ is the scale parameter of Logistic distribution.
The distribution function of Logistic distribution is
$$ \begin{equation*} F(x) =1- \frac{1}{1+e^{\frac{(x-\mu)}{\lambda}}}. \end{equation*} $$
Logistic probabilities using dlogis()
function in R
For continuous probability distribution, density is the value of the probability density function at $x$ (i.e., $f(x)$).
The syntax to compute the probability density function for Logistic distribution using R is
dlogis(x,location, scale)
where
x
: the value(s) of the variable and,location
: location parameter of Logistic distribution,scale
: scale parameter of Logistic distribution.
The dlogis()
function gives the density for given value(s) x
, location
and scale
.
Note: If you do not specify the location
and scale
, R assumes the default value location=0
and scale=1
(which is a standard Logistic distribution).
Numerical Problem for Logistic Distribution
To understand the four functions dlogis()
, plogis()
, qlogis()
and rlogis()
, let us take the following numerical problem.
Logistic Distribution Example
Let random variable $X$ follows a Logistic distribution with $\mu=1$ and $\lambda=2$.
(a) Find the value of the density function at $x=1.5$.
(b) Plot the graph of Logistic probability distribution.
(c) Find the probability that, $X$ is less than 2.
(d) Find the probability that, $X$ is greater than 3.
(e) Find the probability that, $X$ is less than 2.5 but greater than 4.5.
(f) Plot the graph of cumulative Logistic probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.60$?
(h) Simulate 1000 Logistic distributed random variables with $\mu= 1$ and $\lambda = 2$.
Let $X$ denote the daily proportion of major automobile accidents across the United States. Given that $X\sim Logis(\mu=1, \lambda=2)$.
Example 1: How to use dlogis()
function in R?
To find the value of the density function at $x=2.5$ we need to use dlogis()
function.
Let $X\sim Logis(1,2)$.
First let us define the given parameters as
# parameter 1 location
mu <- 1
# parameter 2 scale
lambda <- 2
The probability density function of $X$ is
$$ \begin{aligned} f(x)&= \frac{1}{2}\frac{e^{\frac{(x-1)}{2}}}{\bigg(1+e^{\frac{(x-1)}{2}}\bigg)^2},\\ &\quad\text{for } -\infty \leq x \leq \infty. \end{aligned} $$
For part (a), we need to find the density function at $x=1.5$. That is $f(1.5)$.
First I will show you how to calculate the value of the density function for given value of $x$. Then I will show you how to compute the same using dlogis()
function in R.
(a) The value of the density function at $x=2.5$ is
$$ \begin{aligned} f(2.5)&=\frac{2}{\pi}\cdot \frac{1}{2^2+(2.5-1)^2}\\ &=\frac{2}{\pi}\cdot \frac{1}{2^2+(2.5-1)^2} &= 0.1089475 \end{aligned} $$
The above probability can be calculated using dlogis(2.5,1,2)
function in R.
# Compute Logistic probability
result1 <- dlogis(2.5,mu,lambda)
result1
[1] 0.1089475
Example 2 Visualize Logistic probability distribution
Using dlogis()
function we can compute Logistic distribution probabilities for given x
, location
and scale
. To plot the probability density function of Logistic distribution, we need to create a sequence of x
values and compute the corresponding probabilities.
# create a sequence of x values
x <- seq(-10,14, by=0.02)
## Compute the Logistic pdf for each x
px<-dlogis(x,mu,lambda)
(b) Visualizing Logistic Distribution with dlogis()
function and plot()
function in R:
The probability density function of Logistic distribution with given 1 and 2 can be visualized using plot()
function as follows:
## Plot the Logistic probability dist
plot(x,px,type="l",xlim=c(-10,14),ylim=c(0,max(px)),
lwd=3, col="darkred",ylab="f(x)",
main=expression(paste("PDF of Logis(",
mu,"=1, ",lambda,"=2)")))
abline(v=c(1),lty=2,col="gray")

Logistic cumulative probability using plogis()
function in R
The syntax to compute the cumulative probability distribution function (CDF) for Logistic distribution using R is
plogis(q,location, scale)
where
q
: the value(s) of the variable,location
: first parameter of Logistic distribution,scale
: second parameter of Logistic distribution.
Using this function one can calculate the cumulative distribution function of Logistic distribution for given value(s) of q
(value of the variable x
), location
and scale
.
Example 3: How to use plogis()
function in R?
In the above example, for part (c), we need to find the probability $P(X\leq 3)$.
(c) The probability that $X$ is less than $2$ is
$$ \begin{aligned} P(X \leq 2) &=F(2)\\ &=0.5+\frac{1}{\pi} tan^{-1}\big(\frac{2-1}{2}\big)\\ &=0.5 + \frac{1}{3.1416}tan^{-1}\big(0.5\big)\\ &=0.5 + \frac{1}{3.1416}(0.4636)\\ &= 0.6225 \end{aligned} $$
## Compute cumulative Logistic probability
result2 <- plogis(2,mu,lambda)
result2
[1] 0.6224593
Example 4: How to use plogis()
function in R?
In the above example, for part (d), we need to find the probability $P(X \geq 3)$.
(d) The probability that $X$ is greater than $3$ is
$$ \begin{aligned} P(X > 3) &=1- P(X < 3)\\ &= 1- F(3)\\ &=1-\bigg(0.5+\frac{1}{\pi} tan^{-1}\big(\frac{3-1}{2}\big)\bigg)\\ &=0.5 - \frac{1}{3.1416}tan^{-1}\big(1\big)\\ &=0.5 - \frac{1}{3.1416}(0.7854)\\ &= 0.2689 \end{aligned} $$
To calculate the probability that a random variable $X$ is greater than a given number one can use the option lower.tail=FALSE
in plogis()
function.
Above probability can be calculated easily using plogis()
function with argument lower.tail=FALSE
as
$P(X \geq 3)=$ plogis(3,mu,lambda,lower.tail=FALSE)
or by using complementary event as
$P(X \geq 3) = 1- P(X\leq 3)$= 1- plogis(3,mu,lambda)
# compute cumulative Logistic probabilities
# with lower.tail False
plogis(3,mu,lambda,lower.tail=FALSE)
[1] 0.2689414
# Using complementary event
1-plogis(3,mu,lambda)
[1] 0.2689414
Example 5: How to use plogis()
function in R?
One can also use plogis()
function to calculate the probability that the random variable $X$ is between two values.
(e) The probability that $X$ is between $2.5$ and $4.5$ is
$$ \begin{aligned} P(2.5 \leq X \leq 4.5)&=P(X\leq 4.5)-P(X\leq 2.5)\\ &=F(4.5) -F(2.5)\\ &=\bigg[0.5+\frac{1}{\pi} tan^{-1}\big(\frac{4.5-1}{2}\big)\bigg]-\bigg[0.5+\frac{1}{\pi} tan^{-1}\big(\frac{2.5-1}{2}\big)\bigg]\\ &=\frac{1}{\pi} tan^{-1}\big(1.75\big)-\frac{1}{\pi} tan^{-1}\big(0.75\big)\\ &=\frac{1}{3.1416}(1.0517)-\frac{1}{3.1416}(0.6435)\\ &=0.1728 \end{aligned} $$
The above probability can be calculated using plogis()
function as follows:
a <- plogis(4.5,mu,lambda)
b <- plogis(2.5,mu,lambda)
result3 <- a - b
result3
[1] 0.1727741
Example 6: Visualize the cumulative Logistic probability distribution
Using plogis()
function we can compute Logistic cumulative probabilities (CDF) for given x
, location
and scale
. To plot the CDF of Logistic distribution, we need to create a sequence of x
values and compute the corresponding cumulative probabilities.
# create a sequence of x values
x <- seq(-10,14, by=0.02)
## Compute the Logistic pdf for each x
Fx <- plogis(x,mu,lambda)
(f) Visualizing Logistic Distribution with plogis()
function and plot()
function in R:
The cumulative probability distribution of Logistic distribution with given x
, location
and scale
can be visualized using plot()
function as follows:
## Plot the Logistic probability dist
plot(x,Fx,type="l",xlim=c(-10,14),ylim=c(0,1),
lwd=3, col="darkred",ylab="F(x)",
main=expression(paste("Distribution function of
Logis(",mu,"=1, ",lambda,"=2)")))

Logistic Distribution Quantiles using qlogis()
in R
The syntax to compute the quantiles of Logistic distribution using R is
qlogis(p,location,scale)
where
p
: the value(s) of the probabilities,location
: first parameter of Logistic distribution,scale
: second parameter of Logistic distribution.
The function qlogis(p,location,scale)
gives $100*p^{th}$ quantile of Logistic distribution for given value of p
, location
and scale
.
The $p^{th}$ quantile is the smallest value of Logistic random variable $X$ such that $P(X\leq x) \geq p$.
It is the inverse of plogis()
function. That is, inverse cumulative probability distribution function for Logistic distribution.
Example 7: How to use qlogis()
function in R?
In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.60$. That is we need to find the $60^{th}$ quantile of given Logistic distribution.
mu <- 2
lambda <- 4
prob <- 0.60
# compute the quantile for Logistic dist
qlogis(0.60,mu, lambda)
[1] 3.62186
The $60^{th}$ percentile of given Logistic distribution is 3.6218604.
Visualize the quantiles of Beta Distribution
The quantiles of Beta distribution with given p
, location
and scale
can be visualized using plot()
function as follows:
p <- seq(0,1,by=0.02)
qx <- qlogis(p,mu,lambda)
# Plot the quantiles of Logistic dist
plot(p,qx,type="l",lwd=2,col="darkred",
ylab="quantiles",
main=expression(paste("Quantiles of Logis(",
mu,"=1, ",lambda,"=2)")))

Simulating Logistic random variable using rlogis()
function in R
The general R function to generate random numbers from Logistic distribution is
rlogis(n,location,scale)
where,
n
: the sample observations,location
: first parameter of Logistic distribution,scale
: second parameter of Logistic distribution.
The function rlogis(n,location,scale)
generates n
random numbers from Logistic distribution with given location
and scale
.
Example 8: How to use rlogis()
function in R?
In part (h), we need to generate 1000 random numbers from Logistic distribution with given $location = 2$ and $scale=4$.
(h) We can use rlogis(1000,mu,lambda)
function to generate random numbers from Logistic distribution.
## initialize sample size to generate
n <- 1000
# Simulate 1000 values From Logistic dist
x_sim <- rlogis(n,mu,lambda)
The below graphs shows the density of the simulated random variables from Logistic Distribution.
## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main=expression(paste("Simulated Data from Logis(",
mu,"=1, ",lambda,"=2)")))

If you use same function again, R will generate another set of random numbers from $Logis(2,4)$.
# Simulate 1000 values From Logistic dist
x_sim_2 <- rlogis(n,mu,lambda)
## Plot the simulated data
plot(density(x_sim_2),xlab="Simulated x",ylab="density",
lwd=5,col="blue",
main=expression(paste("Simulated Data from Logis(",
mu,"=1, ",lambda,"=2)")))

For the simulation purpose to reproduce same set of random numbers, one can use set.seed()
function.
# set seed for reproducibility
set.seed(1457)
# Simulate 1000 values From Logistic dist
x_sim_3 <- rlogis(n,mu,lambda)
## Plot the simulated data
plot(density(x_sim_3),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main=expression(paste("Simulated Data from Logis(",
mu,"=1, ",lambda,"=2)")))

set.seed(1457)
# Simulate 1000 values From Logistic dist
x_sim_4 <- rlogis(n,mu,lambda)
## Plot the simulated data
plot(density(x_sim_4),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main=expression(paste("Simulated Data from Logis(",
mu,"=1, ",lambda,"=2)")))

Since we have used set.seed(1457)
function, R will generate the same set of Logistic distributed random numbers.
hist(x_sim_4,breaks = 50,col="red4",
main=expression(paste("Histogram Logis(",
mu,"=1, ",lambda,"=2)")))

To learn more about other discrete and continuous probability distributions using R, go through the following tutorials:
Discrete Distributions Using R
Binomial distribution in R
Poisson distribution in R
Geometric distribution in R
Negative Binomial distribution in R
Hypergeometric distribution in R
Continuous Distributions Using R
Uniform distribution in R
Exponential distribution in R
Normal distribution in R
Log-normal distribution in R
Gamma distribution in R
Beta distribution in R
Cauchy distribution in R
Laplace distribution in R
Weibull distribution in R
Endnote
In this tutorial, you learned about how to compute the probabilities, cumulative probabilities and quantiles of Logistic distribution in R programming. You also learned about how to simulate a Logistic distribution using R programming.
To learn more about R code for discrete and continuous probability distributions, please refer to the following tutorials:
Probability Distributions using R
Let me know in the comments below, if you have any questions on Logistic Distribution using R and your thought on this article.