Contents

- 1 Logistic Distribution probabilities using R
- 2 Logistic Distribution
- 3 Logistic probabilities using dlogis() function in R
- 4 Numerical Problem for Logistic Distribution
- 5 Logistic cumulative probability using plogis() function in R
- 6 Logistic Distribution Quantiles using qlogis() in R
- 7 Simulating Logistic random variable using rlogis() function in R
- 8 Endnote

## 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.