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# Exponential Distribution probabilities using R

## Exponential Distribution probabilities using R

In this tutorial, you will learn about how to use dexp(), pexp(), qexp() and rexp() functions in R programming language to compute the individual probabilities, cumulative probabilities, quantiles and to generate random sample for Exponential distribution.

Before we discuss R functions for Exponential distribution, let us see what is Exponential distribution.

## Exponential Distribution

Exponential distribution distribution is a continuous type probability distribution.

Exponential distribution is often used to model the lifetime of electric components. It is routinely used as a survival distribution in survival analysis and reliability analysis.

Let $X\sim Exp(\lambda)$. Then the probability distribution of $X$ is

 \begin{aligned} f(x)&= \begin{cases} \lambda e^{-\lambda x}, & x > 0;\lambda> 0; \\ 0, & Otherwise. \end{cases} \end{aligned}

where $\lambda$ is the scale parameter (also known as rate) of Exponential distribution.

## Exponential probabilities using dexp() 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 Exponential distribution using R is

dexp(x,rate=1)

where

• x : the value(s) of the variable and,
• rate : rate parameter of exponential distribution.

Note: If you do not specify the rate, R assumes the default value rate=1 (which is a standard exponential distribution).

The dexp() function gives the density for given value(s) x and rate.

## Numerical Problem for Exponential Distribution

To understand the four functions dexp(), pexp(), qexp() and rexp(), let us take the following numerical problem.

### Exponential Distribution Example

The time (in hours) required to repair a machine is an exponential distributed random variable with paramter $\lambda=1/2$.

(a) Find the value of the density function at $x=2.5$.
(b) Plot the graph of Exponential probability distribution.
(c) Find the probability that a repair time takes at most 3 hours.
(d) Find the probability that a repair time exceeds 4 hours.
(e) Find the probability that a repair time takes between 2 to 4 hours.
(f) Plot the graph of cumulative Exponential probabilities.
(g) What is the value of $c$, if $P(X\leq c) \geq 0.50$?
(h) Simulate 1000 Exponential distributed random variables with $\lambda= 1/2$.

Let $X$ denote the time (in hours) required to repair a machine. Given that $X\sim Exp(\lambda=1/2)$.

### Example 1: How to use  dexp() function in R?

To find the value of the density function at $x=2.5$ we need to use  dexp() function.

First let us define the given parameters as

# scale parameter
lambda <- 1/2

The probability density function of $X$ is

 \begin{aligned} f(x)&= \frac{1}{2} e^{-x/2},\\ &\quad\text{for } x \geq 0. \end{aligned}

For part (a), we need to find the density function at $x=2.5$. That is $f(2.5)$.

(a) The value of the density function at $x=2.5$ is

 \begin{aligned} f(2.5)&= \frac{1}{2}\times e^{-2.5/2}\\ &=2.5\times e^{-1.25}\\ &= 0.1432524 \end{aligned}

The above probability can be calculated using dexp(2.5,rate=0.5) function in R.

# Compute Exponential probability
result1 <- dexp(2.5,rate=lambda)
result1
[1] 0.1432524

### Example 2 Visualize Exponential probability distribution

Using dexp() function we can compute Exponential distribution probabilities for given x and rate. To plot the probability density function of Exponential distribution, we need to create a sequence of x values and compute the corresponding probabilities.

# create a sequence of x values
x <- seq(0,5, by=0.02)
## Compute the Exponential pdf for each x
px<- dexp(x,rate=lambda)

(b) Visualizing Exponential Distribution with dexp() function and plot() function in R:

The probability density function of Exponential distribution with given 0.5 can be visualized using plot() function as follows:

## Plot the Exponential probability dist
plot(x,px,type="l",xlim=c(0,5),ylim=c(0,max(px)),
lwd=3, col="darkred",ylab="f(x)")
title("PDF of Exponential (lambda = 1/2)")

## Exponential cumulative probability using pexp() function in R

The syntax to compute the cumulative probability distribution function (CDF) for Exponential distribution using R is

pexp(q, rate=1)

where

• q : the value(s) of the variable,
• rate : scale parameter of exponential distribution.

Using this function one can calculate the cumulative distribution function of Exponential distribution for given value(s) of q (value of the variable x), rate.

### Example 3: How to use pexp() function in R?

In the above example, for part (c), we need to find the probability $P(X\leq 3)$.

(c) The probability that a repair time takes at most 3 hours is

 \begin{aligned} P(X\leq 3) &=\int_0^{3} f(x)\; dx. \end{aligned}

## Compute cumulative Exponential probability
result2 <- pexp(3,rate=lambda)
result2
[1] 0.7768698

### Example 4: How to use pexp() function in R?

In the above example, for part (d), we need to find the probability $P(X \geq 4)$.

To calculate the probability that a random variable $X$ is greater than a given number, one can use the option lower.tail=FALSE in pexp() function.

Above probability can be calculated easily using pexp() function with argument lower.tail=FALSE as

$P(X \geq 4) =\int_{4}^\infty f(x)\; dx$= pexp(4,rate=lambda,lower.tail=FALSE)

or by using complementary event as

$P(X \geq 4) = 1- P(X\leq 4)$= 1- pexp(4,rate=lambda)

# compute cumulative Exponential probabilities
# with lower.tail False
pexp(4,rate=lambda,lower.tail=FALSE)
[1] 0.1353353

(d) The probability that a repair time exceeds 4 hours is

 \begin{aligned} P(X\geq 4) &=\int_{4}^\infty f(x)\; dx\\ &=0.1353353. \end{aligned}

# Using complementary event
1-pexp(4,rate=lambda)
[1] 0.1353353

### Example 5: How to use pexp() function in R?

One can also use pexp() function to calculate the probability that the random variable $X$ is between two values.

(e) The probability that a repair time takes between 2 to 4 hours can be written as $P(2 < X < 4)$.

 \begin{aligned} P(2 < X < 4) &= P(X< 4) -P(X < 2)\\ &= 0.8646647 - 0.6321206\\ &= 0.2325442 \end{aligned}

The above probability can be calculated using pexp() function as follows:

a <- pexp(4,rate=lambda)
b <- pexp(2,rate=lambda)
result3 <- a - b
result3
[1] 0.2325442

### Example 6: Visualize the cumulative Exponential probability distribution

Using pexp() function we can compute Exponential cumulative probabilities (CDF) for given x and rate. To plot the CDF of Exponential distribution, we need to create a sequence of x values and compute the corresponding cumulative probabilities.

# create a sequence of x values
x <- seq(0,5, by=0.02)
## Compute the Exponential pdf for each x
Fx <- pexp(x,rate=lambda)

(f) Visualizing Exponential Distribution with pexp() function and plot() function in R:

The cumulative probability distribution of Exponential distribution with given x and rate can be visualized using plot() function as follows:

## Plot the Exponential  probability dist
plot(x,Fx,type="l",xlim=c(0,5),ylim=c(0,1),
lwd=3, col="darkred",ylab="F(x)")
title("CDF of Exp (lambda= 1/2)")

## Exponential Distribution Quantiles using qexp() in R

The syntax to compute the quantiles of Exponential distribution using R is

qexp(p,rate=1)

where

• p : the value(s) of the probabilities,
• rate =1 : scale parameter of exponential distribution.

The function qexp(p,rate=1) gives $100*p^{th}$ quantile of Exponential distribution for given value of p, and rate.

The $p^{th}$ quantile is the smallest value of Exponential random variable $X$ such that $P(X\leq x) \geq p$.

It is the inverse of pexp() function. That is, inverse cumulative probability distribution function for Exponential distribution.

### Example 7: How to use qexp() function in R?

In part (g), we need to find the value of $c$ such a that $P(X\leq c) \geq 0.50$. That is we need to find the $50^{th}$ quantile of given Exponential distribution.

lambda <- 1/2
prob <- 0.50
# compute the quantile for Exponential  dist
qexp(0.50,rate=lambda)
[1] 1.386294

The $50^{th}$ percentile of given Exponential distribution is 1.3862944.

### Visualize the quantiles of exponential Distribution

The quantiles of exponential distribution with given p and rate=lambda can be visualized using plot() function as follows:

p <- seq(0,1,by=0.02)
qx <- qexp(p,rate=lambda)
# Plot the Quantiles of Exponential  dist
plot(p,qx,type="l",lwd=2,col="darkred",
ylab="quantiles",
main="Quantiles of Exponential(lambda=1/2)")

## Simulating Exponential random variable using rexp() function in R

The general R function to generate random numbers from Exponential distribution is

rexp(n,rate=1)

where,

• n : the sample observations,
• rate : scale parameter of exponential distribution.

The function rexp(n,rate) generates n random numbers from Exponential distribution with given rate.

### Example 8: How to use rexp() function in R?

In part (h), we need to generate 1000 random numbers from Exponential distribution with given $rate = 0.5$.

(h) We can use rexp(1000,rate) function to generate random numbers from Exponential distribution.

## initialize sample size to generate
n <- 1000
# Simulate 1000 values From Exponential  dist
x_sim <- rexp(n,rate=lambda)

The below graphs shows the density of the simulated random variables from Exponential Distribution.

## Plot the simulated data
plot(density(x_sim),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Exponential(lambda=1/2) dist")

If you use same function again, R will generate another set of random numbers from $Exp(0.5)$.

# Simulate 1000 values From Exponential  dist
x_sim_2 <- rexp(n,rate=lambda)
## Plot the simulated data
plot(density(x_sim_2),xlab="Simulated x",ylab="density",
lwd=5,col="blue",
main="Simulated data from Exp(lambda=1/2) dist")

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 Exponential  dist
x_sim_3 <- rexp(n,rate=lambda)
## Plot the simulated data
plot(density(x_sim_3),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Exp(lambda=1/2) dist")
set.seed(1457)
# Simulate 1000 values From Exponential  dist
x_sim_4 <- rexp(n,rate=lambda)
## Plot the simulated data
plot(density(x_sim_4),xlab="Simulated x",ylab="density",
lwd=5,col="darkred",
main="Simulated data from Exp(lambda=1/2) dist")

Since we have used set.seed(1457) function, R will generate the same set of Exponential distributed random numbers.

hist(x_sim_4,breaks = 30)

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 Exponential distribution in R programming. You also learned about how to simulate a Exponential distribution using R programming.