Log-normal Distribution Probabilities using R
In this tutorial, you will learn about how to use dlnorm(), plnorm(), qlnorm() and rlnorm() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Log-normal distribution.
Before we discuss R functions for Log-normal distribution, let us see what is Log-normal distribution.
Log-normal Distribution
Log-normal distribution distribution is a continuous type probability distribution. Log-normal distribution is commonly used in finance to analyze the stock prices.
The continuous random variable $X$ has a Log-Normal Distribution if the random variable $Y=\ln (X)$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$. The probability density function of $X$ is
$$ \begin{align*} f(x;\mu,\sigma) &= \begin{cases} \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}(\ln x -\mu)^2}, & x\geq 0; \\ 0, & x < 0. \end{cases} \end{align*} $$
$\mu$is location parameter$\sigma$is scale parameter
where $e= 2.71828...$ and $\pi = 3.1425926...$.
In notation it can be written as $X\sim LN(\mu,\sigma^2)$.
If $X\sim LN(\mu,\sigma^2)$ then $\ln X \sim N(\mu, \sigma^2)$.
Read more about the theory and results of Log-normal distribution here.
Log-normal probabilities using dlnorm() 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 Log-normal distribution using R is
dlnorm(x,meanlog=0, sdlog = 1)where
x: the value(s) of the variable and,meanlog: mean of the distribution on log scale,sdlog: standard deviation of the distribution on log scale.
Note: If you do not specify the meanlog and sdlog, R will take the default value of meanlog as 0 and sdlog as 1.
The dlnorm() function gives the density for given value(s) x, meanlog and sdlog.
Numerical Problem for Log-normal Distribution
To understand the four functions dlnorm(), plnorm(), qlnorm() and rlnorm(), let us take the following numerical problem.
Log-normal Distribution Example
The life-time (in days) of certain electronic component that operates in a high-temperature environment is log-normally distributed with parameters $\mu=1.2$ and $\sigma=0.5$.
(a) Find the value of the density function at $x=3.5$.
(b) Plot the graph of Log-normal probability distribution.
(c) Find the probability that the component works till 4 days.
(d) Find the probability that the component works more than 5 days.
(e) Find the probability that the component works between 3 and 5 days.
(f) Plot the graph of cumulative Log-normal probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.80$?
(h) Simulate 1000 Log-normal distributed random variables with $\mu= 1.2$ and $\sigma = 0.5$.
Let $X$ denote the life-time (in days) of certain electronic components that operates in a high-temperature environment. Given that $X\sim LN(1.2, 0.5^2)$. That is $\mu = 1.2$ and $\sigma = 0.5$.
Then $\ln(X)\sim N(1.2,0.25)$ distribution.
Example 1: How to use dlnorm() function in R?
To find the value of the density function at $x=3.5$ we need to use dlnorm() function.
First let us define the given parameters as
# mean log of distribution
mu <- 1.2
# sd log of distribution
sigma <- 0.5The probability density function of $X$ is
$$ \begin{align*} f(x;\mu,\sigma) &= \begin{cases} \frac{1}{0.5 x\sqrt{2\pi} }e^{-\frac{1}{2}\big(\frac{\ln x -1.2}{0.5}\big)^2}, & x\geq 0; \\ 0, & x < 0. \end{cases} \end{align*} $$
For part (a), we need to find the density function at $x=3.5$. That is $f(3.5)$.
(a) The value of the density function at $x=3.5$ is
$$ \begin{aligned} f(3.5)&=\frac{1}{0.5\times 3.5\times \sqrt{2\pi}}e^{-\frac{1}{2}\big(\frac{\ln(3.5)-1.2}{0.5}\big)^2}\\ &= 0.2267013 \end{aligned} $$
The above probability can be calculated using dlnorm(3.5,meanlog=1.2,sdlog=0.5) function in R.
# Compute Log-normal probability
result1 <- dlnorm(3.5,meanlog=mu,sdlog=sigma)
result1[1] 0.2267013Example 2 Visualize Log-normal probability distribution
Using dlnorm() function we can compute Log-normal distribution probabilities for given x, meanlog and sdlog. To plot the probability density function of Log-normal distribution, we need to create a sequence of x values and compute the corresponding probabilities.
# create a sequence of x values
x <- seq(0,12, by=0.01)
## Compute the Log-normal pdf for each x
px<- dlnorm(x,meanlog=mu,sdlog=sigma)(b) Visualizing Log-normal Distribution with dlnorm() function and plot() function in R:
The probability density function of Log-normal distribution with given $\mu=1.2$ and $\sigma=0.5$ can be visualized using plot() function as follows:
## Plot the Log-normal probability dist
plot(x,px,type="l",xlim=c(0,12),ylim=c(0,max(px)),
lwd=3, col="blue4",ylab="f(x)",
main=expression(paste("PDF of Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
Log-normal cumulative probability using plnorm() function in R
The syntax to compute the cumulative probability distribution function (CDF) for Log-normal distribution using R is
plnorm(q,meanlog=0, sdlog=1)where
q: the value(s) of the variable,meanlog: mean of the distribution on log scale,sdlog: standard deviation of the distribution on log scale.
Using this function one can calculate the cumulative distribution function of Log-normal distribution for given value(s) of q (value of the variable x), meanlog and sdlog.
Example 3: How to use plnorm() function in R?
In the above example, for part (c), we need to find the probability $P(X\leq 4)$.
(c) The probability that the component works till 4 days is $P(X < 4)$.
The $Z$ score that corresponds to $4$ is
$$ \begin{aligned} z&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(4)-1.2}{0.5}\\ &\approx0.37 \end{aligned} $$
Thus the probability that the component works till 4 days is
$$ \begin{aligned} P(X < 4) &=P(\ln(X) < \ln(4))\\ &=P(Z < 0.37)\\ &=0.6443088 \end{aligned} $$
## Compute cumulative Log-normal probability
result2 <- plnorm(4,meanlog=mu,sdlog=sigma)
result2[1] 0.6452727Example 4: How to use plnorm() function in R?
In the above example, for part (d), we need to find the probability $P(X \geq 5)$.
(d) The probability that the component works more than 5 days is $P(X > 5)$.
The $Z$ score that corresponds to $5$ is
$$ \begin{aligned} z&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(5)-1.2}{0.5}\\ &\approx0.82 \end{aligned} $$
The probability that the component works more than 5 days is
$$ \begin{aligned} P(X > 5) &=1-P(X < 5)\\ &= 1-P(\ln X < \ln (5))\\ &= 1-P(Z < 0.82)\\ &=1-0.7938919\\ &=0.2061081 \end{aligned} $$
To calculate the probability that a random variable $X$ is greater than a given number, one can use the option lower.tail=FALSE in plnorm() function.
Above probability can be calculated easily using plnorm() function with argument lower.tail=FALSE as
$P(X \geq 5)$= plnorm(5,meanlog=1.2,sdlog=0.5,lower.tail=FALSE)
or by using complementary event as
$P(X \geq 5) = 1- P(X\leq 5)$= 1- plnorm(5,meanlog=1.2,sdlog=0.5)
# compute cumulative Log-normal probabilities
# with lower.tail False
plnorm(5,meanlog=mu,sdlog=sigma,lower.tail=FALSE)[1] 0.2064286# Using complementary event
1-plnorm(5,meanlog=mu,sdlog=sigma)[1] 0.2064286Example 5: How to use plnorm() function in R?
One can also use plnorm() function to calculate the probability that the random variable $X$ is between two values.
(e) The probability that the component works between 3 and 5 days is $P(3 < X < 5)$.
The Z score that corresponds to $3$ and $5$ are respectively
$$ \begin{aligned} z_1&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(3)-1.2}{0.5}\\ &\approx-0.2 \end{aligned} $$
and
$$ \begin{aligned} z_2&=\dfrac{\ln(X)-\mu}{\sigma}\\ &=\dfrac{\ln(5)-1.2}{0.5}\\ &\approx0.82 \end{aligned} $$
The probability that the component works between 3 and 5 days is
$$ \begin{aligned} P(3 \leq X\leq 5) &=P(\ln (3) \leq \ln X\leq \ln(5))\\ &=P(-0.2\leq Z\leq 0.82)\\ &= P(Z < 0.82) -P( Z < -0.2)\\ &=0.7938919-0.4207403\\ &= 0.3731517 \end{aligned} $$
The above probability can be calculated using plnorm() function as follows:
a <- plnorm(5,meanlog=mu,sdlog=sigma)
b <- plnorm(3,meanlog=mu,sdlog=sigma)
result3 <- a - b
result3[1] 0.3739161Example 6: Visualize the cumulative Log-normal probability distribution
Using plnorm() function we can compute Log-normal cumulative probabilities (CDF) for given x, meanlog and sdlog. To plot the CDF of Log-normal distribution, we need to create a sequence of x values and compute the corresponding cumulative probabilities.
# create a sequence of x values
x <- seq(0,12, by=0.01)
## Compute the Log-normal pdf for each x
Fx <- plnorm(x,meanlog=mu,sdlog=sigma)(f) Visualizing Log-normal Distribution with plnorm() function and plot() function in R:
The cumulative probability distribution of Log-normal distribution with given x, meanlog and sdlog can be visualized using plot() function as follows:
## Plot the Log-normal probability dist
plot(x,Fx,type="l",xlim=c(0,12),ylim=c(0,1),
lwd=3, col="blue4",ylab="F(x)",
main=expression(paste("CDF of Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
Log-normal Distribution Quantiles using qlnorm() in R
The syntax to compute the quantiles of Log-normal distribution using R is
qlnorm(p,meanlog=0,sdlog=1)where
p: the value(s) of the probabilities,meanlog: mean of the distribution on log scale,sdlog: standard deviation of the distribution on log scale.
The function qlnorm(p,meanlog,sdlog) gives $100*p^{th}$ quantile of Log-normal distribution for given value of p, meanlog and sdlog.
The $p^{th}$ quantile is the smallest value of Log-normal random variable $X$ such that $P(X\leq x) \geq p$.
It is the inverse of plnorm() function. That is, inverse cumulative probability distribution function for Log-normal distribution.
Example 7: How to use qlnorm() function in R?
In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.80$. That is we need to find the $80^{th}$ quantile of given Log-normal distribution.
mu <- 1.2
sigma <- 0.5
prob <- 0.80# compute the quantile for Log-normal dist
qlnorm(0.80,meanlog=mu, sdlog=sigma)[1] 5.057188The $80^{th}$ percentile of given Log-normal distribution is 5.0571881.
Visualize the quantiles of Log-normal Distribution
The quantiles of Log-normal distribution with given p, meanlog=mu and sdlog=sigma can be visualized using plot() function as follows:
p <- seq(0,1,by=0.01)
qx <- qlnorm(p,meanlog=mu,sdlog=sigma)
# Plot the Quantiles of Log-normal dist
plot(p,qx,type="l",lwd=2,col="blue4",
ylab="quantiles",
main=expression(paste("Quantiles of Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
Simulating Log-normal random variable using rlnorm() function in R
The general R function to generate random numbers from Log-normal distribution is
rlnorm(n,meanlog=0,sdlog=1)where,
n: the sample observations,meanlog: mean of the distribution on log scale,sdlog: standard deviation of the distribution on log scale.
The function rlnorm(n,meanlog,sdlog) generates n random numbers from Log-normal distribution with given meanlog and sdlog.
Example 8: How to use rlnorm() function in R?
In part (h), we need to generate 1000 random numbers from Log-normal distribution with given $meanlog = 1.2$ and $sdlog=0.5$.
(h) We can use rlnorm(1000,meanlog,sdlog) function to generate random numbers from Log-normal distribution.
## initialize sample size to generate
n <- 1000
# Simulate 1000 values From Log-normal dist
x_sim <- rlnorm(n,meanlog=mu,sdlog=sigma)The below graphs shows the density of the simulated random variables from Log-normal Distribution.
## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main=expression(paste("Simulated data from Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
If you use same function again, R will generate another set of random numbers from $LN(1.2,0.5^2)$.
# Simulate 1000 values From Log-normal dist
x_sim_2 <- rlnorm(n,meanlog=mu,sdlog=sigma)## Plot the simulated data
plot(density(x_sim_2),xlab="Simulated x",ylab="density",
lwd=5,col="blue",
main=expression(paste("Simulated data from Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
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 Log-normal dist
x_sim_3 <- rlnorm(n,meanlog=mu,sdlog=sigma)## Plot the simulated data
plot(density(x_sim_3),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main=expression(paste("Simulated data from Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
set.seed(1457)
# Simulate 1000 values From Log-normal dist
x_sim_4 <- rlnorm(n,meanlog=mu,sdlog=sigma)## Plot the simulated data
plot(density(x_sim_4),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main=expression(paste("Simulated data from Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))Since we have used set.seed(1457) function, R will generate the same set of Log-normal distributed random numbers.
hist(x_sim_4,breaks = 30,col="blue4",
main=expression(paste("Histogram of Simulated data from Log-normal with ",
mu,"=1.2 and ",sigma,"=0.5")))
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
Beta distribution in R
Gamma distribution in R
Cauchy distribution in R
Laplace distribution in R
Logistic distribution in R
Weibull distribution in R
Endnote
In this tutorial, you learned about how to compute the probabilities, cumulative probabilities and quantiles of Log-normal distribution in R programming. You also learned about how to simulate a Log-normal 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 Log-normal Distribution using R and your thought on this article.
