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# Log-normal Distribution Probabilities using R

## 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 $\ln X \sim N(\mu, \sigma^2)$.

## 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 <- 0.5

The 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
 0.2267013

### Example 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
 0.6452727

### Example 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)
 0.2064286
# Using complementary event
1-plnorm(5,meanlog=mu,sdlog=sigma)
 0.2064286

### Example 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
 0.3739161

### Example 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)
 5.057188

The $80^{th}$ 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

Continuous Distributions Using 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.