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How to Generate Random Number from Standard Distributions in R

Random number generation is a crucial aspect of statistical simulations and data analysis. R languauge is a powerful statistical computing language, generating random numbers from standard distributions is a routine task.

Standard Distribution

A “standard” probability distribution refers to a specific type of probability distribution that has been standardized or normalized in a way that allows for easy comparison or analysis. For example, in the case of the normal distribution, a standard normal distribution has a mean (average) of 0 and a standard deviation of 1.

Standard distributions in R language basically five types like – Normal, Uniform ,Exponential,Binomial and Poisson.

Normal Distribution (Gaussian Distribution)

Normal Distribution is a continuous probability distribution that is symmetric and bell-shaped. It is characterized by its mean (μ) and standard deviation (σ). The standard normal distribution is a specific form of the normal distribution with μ = 0 and σ = 1.

f(x;\upsilon ,\sigma ) = \frac{1}{\sqrt{2\pi } \sigma }e ^\frac{{-(x-\mu )^{2}}}{2\sigma ^{2}}

Where:

  • -∞ < x < ∞

Uniform Distribution

Uniform distribution is a probability distribution where all outcomes are equally likely. For example, rolling a fair six-sided die has a uniform distribution because each side has an equal chance of landing face up.

f(x) = \frac{1}{b-a}

Where:

  • a ≤ x ≤ b

Exponential Distribution

Exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process. It is often used in situations involving the modeling of time intervals between events.

f(x;\lambda ) = \lambda e^{-\lambda x}

Where:

  • x ≥ 0

Binomial Distribution

Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials (experiments with two possible outcomes, typically success or failure).

P(X=x) =\binom{n}{x}p^{x}q^{n-x}

Where:

  • n: number of trials
  • p: probability of success
  • q: probability of failure (p=1-q)
  • X: number of successes in n trials

Poisson Distribution

Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence.

P(x;\lambda ) = \frac{e^{-\lambda }\lambda ^{x}}{x!}

Where:

  • x = 0,1,2,3…

To get started, follow these necessary steps:-

Install and Load R Packages

R

install.packages('stats')
library(stats)

First ensure you have R installed. Load the necessary packages using the “install.packages()” and “library()”commands for specific distributions like ‘stats’ for Normal distribution, Uniform distribution, and Exponential distribution.

Generate Random Numbers

Normal Distribution

Syntax:

rnorm(n, mean, sd)

where

  • n is the number of observations,
  • mean is the mean of the distribution, and
  • sd is the standard deviation.

R

print(rnorm(100, mean = 0, sd = 1))

Output:

[1]  0.758942529 -0.453723834  0.912949296  0.717898789  0.791543559 -0.550075758  0.870248179 -0.466166451
  [9] -0.895218780 -0.337561284  0.717595419  0.133850963  0.681790129  1.255716716  0.108504918 -1.370314518
 [17] -1.191296124 -0.073101164 -1.737569385  0.183472925  0.880325791  0.040857115  1.241827081  0.196701663
 [25]  2.040058850  1.526720480 -0.141765992  0.301710157 -0.328031397  0.764369404  0.258604662 -0.198845302
 [33]  0.003747265 -0.107614661 -1.621863906 -0.660088485 -1.459307096 -0.564311991  1.316207435 -0.062338832
 [41]  0.443187066  0.172675253  0.261057092  2.154863589 -1.040186863  0.922992861 -1.040073052 -1.925036595
 [49] -0.764804674  1.878344170 -1.779070294  1.273924001 -0.343380947  0.069711619  1.106903432 -0.155138403
 [57]  0.832997857  0.383870496 -0.141183939  0.559139232 -0.344832450  2.457222849  1.108045131  1.865635601
 [65]  1.243450462  0.563266845  0.806369221 -1.309192943 -1.290963436  0.700363377  0.658688551 -0.188743862
 [73]  0.245341003 -0.698437649 -1.414918391  0.217500414  0.418729014  0.735709031 -0.549660892  0.874029993
 [81]  0.628098569 -0.584199721 -0.380086985  0.665529536  0.476385817  0.663277136 -1.177093758  0.237022601
 [89]  0.548552073 -0.978999404 -0.578105278  0.110176802  0.855077220 -0.711799781 -0.710650271  1.163810670
 [97]  0.901417598 -0.788738636 -0.376138918 -0.141028442

We generated 100 random numbers from normal distribution

Uniform Distribution

Syntax:

runif(n, min, max)

where:

  • n is the number of observations,
  • min is the lower bound, and
  • max is the upper bound.

R

random_uniform <- runif(50, min = 0, max = 1)
print(random_uniform)

Output:

 [1] 0.50676992 0.69605063 0.29005553 0.34869535 0.90484370 0.85910707 0.70495214 0.79182671 0.01119734 0.40899106
[11] 0.81495474 0.05490031 0.74002386 0.07122289 0.78516947 0.85784466 0.42315433 0.62709236 0.80041710 0.91264925
[21] 0.94827721 0.22207699 0.38470803 0.09788792 0.15572991 0.73780769 0.11070697 0.68107977 0.05804994 0.73271743
[31] 0.04849818 0.60440056 0.15566900 0.51244807 0.79079711 0.77810000 0.61108455 0.07438912 0.05738890 0.52954094
[41] 0.58058220 0.79062618 0.04315235 0.36860145 0.57379876 0.79154094 0.74178902 0.90239004 0.49193354 0.49266193

Generating 50 random numbers from a Uniform distribution between 0 and 1

Exponential Distribution:

Syntax:

rexp(n, rate)

where:

  • rate: represents the shape.
  • n: Specify sample size

R

random_exponential <- rexp(75, rate = 0.5)
print(random_exponential)

Output:

 [1] 0.599724626 2.372824108 1.814010210 4.975601312 6.970033478 0.960039768 1.507752137 1.145074962 1.181036658
[10] 0.219497249 5.237886768 1.807545520 1.423358488 2.338900728 4.136314439 0.654886615 2.485255983 0.481323684
[19] 5.305394348 2.327880467 1.278035887 1.140645239 0.712455931 1.401187526 0.544935975 3.360479098 0.758573133
[28] 0.796037794 0.206771806 3.823335115 1.081069200 0.968634650 5.143709499 3.140314169 0.070044713 6.066294968
[37] 0.344958772 0.925376899 0.534956788 1.658513910 0.805072598 0.167028401 3.483325886 1.798591476 4.677147258
[46] 0.675503581 5.517120767 1.864244714 1.922077855 0.505001211 0.179893075 0.859651250 2.106099440 0.656650158
[55] 2.977580065 2.396910693 0.332511657 0.368489315 0.479358466 0.001085686 1.294693221 3.342880194 1.212357279
[64] 0.283231467 3.109467693 1.061213733 0.047089372 0.978645205 2.109215490 1.196788696 0.355254514 3.545218974
[73] 2.073593198 2.528796159 0.131268121

We are generating 75 random numbers from an Exponential distribution with rate parameter 0.5

Binomial Distribution

Syntax:

rbinom(n, size, prob)

Where:

  • n is number of observations.
  • size is the number of trials.
  • prob is the probability of success of each trial.

The “rbinom” function generates random numbers following a binomial distribution.

R

#  Simulating coin flips
# Probability of success (getting heads)
p_success <- 0.5
 
# Number of trials (coin flips)
n_trials <- 10
 
# Generate random numbers following a binomial distribution
random_binomial <- rbinom(1, n = n_trials, prob = p_success)
print(random_binomial)

Output :

[1] 0 0 0 1 1 0 0 0 0 0

The output, which is stored in random_binomial, represents the number of times you obtained heads (success) in 10 simulated coin flips. Since you are generating only one random value, it will be either 0 (no heads) or 1 (one head) based on the probability of success. For a fair coin (p_success = 0.5), the output will vary between 0 and 1 with roughly equal probability.

Poisson Distribution

Syntax:

rpois(n, lambda)

Where:

  • n: number of random variables to generate
  • lambda: average rate of success

R

# Example: Simulating call arrivals in a call center
# Average rate of calls per hour
lambda <- 10
 
# Time period (in hours)
time_period <- 1
 
# Generate random numbers following a Poisson distribution
random_poisson <- rpois(10, lambda = lambda * time_period)
 
print(random_poisson)

Output:

 [1]  9 10  8 10  9 13  6 14 12 17

The average rate of calls per hour (lambda) is set at 10. The “rpois” function generates 10 random numbers following a Poisson distribution.


 


Reffered: https://www.geeksforgeeks.org

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