The article is a discussion about Probability Distributions in R Language. Master probability distributions in R with this comprehensive guide! Learn how to work with Normal, Binomial, Poisson, Exponential, and other key distributions using built-in R functions. Discover practical examples for PDFs, CDFs, random sampling, statistical tests, and data fitting. Perfect for data scientists, statisticians, and R programmers!
Table of Contents
Probability distributions are the foundation of statistical analysis and data modeling. Whether you are performing hypothesis testing, simulations, or predictive modeling, understanding how to use probability distributions in R is essential. We often make probabilistic statements when working with statistical Probability Distributions. We want to know four things:
- The density (PDF) at a particular value,
- The distribution (CDF) at a particular probability,
- The quantile value corresponding to a particular probability, and
- A random draw of values from a particular distribution.
Probability Distributions in R Language
R language has plenty of functions for obtaining density, distribution, quantile, and random numbers and variables.
Consider a random variable $X$ which is $N(\mu = 2, \sigma^2 = 16)$. We want to:
1) Calculate the value of PDF at $x=3$ (that is, the height of the curve at $x=3$)
dnorm(x = 3, mean = 2, sd = sqrt(16) ) dnorm(x = 3, mean = 2, sd = 4) dnorm(x = 3, 2, 4)
2) Calculate the value of the CDF at $x=3$ (that is, $P(X\le 3)$)
pnorm(q = 3, m = 2, sd = 4)
3) Calculate the quantile for probability 0.975
qnorm(p = 0.975, m = 2, sd = 4)
4) Generate a random sample of size $n = 10$
rnorm(n = 10, m = 2, sd = 5)
There are many probability distributions available in the R Language. The following is the commonly used probability distributions n R Language.
| Binomial | dbinom( ) | qbinom( ) | pbinom( ) | rbinom( ) |
| t | dt( ) | qt( ) | pt( ) | rt( ) |
| Poisson | dpois( ) | qpois( ) | ppois( ) | rpois( ) |
| f | df( ) | qf( ) | pf( ) | rf( ) |
| Chi-Square | dchisq( ) | qchisq( ) | pchisq( ) | rchisq() |
Observe that a prefix (d, q, p, and r) is added for each distribution.
| Distribution | Distribution Name in R | Parameters |
| Binomial | binom | n = Number of trials, and p= probability of success for one trial |
| Geometric | geom | p=probability of success for one trial |
| Poisson | pois | lambda = mean |
| Beta | beta | shape1, shape2 |
| Chi-Square | chisq | df=degrees of freedom |
| F | f | df1, df2 degrees of freedom |
| Logistic | logis | location, scale |
| normal | norm | mean, sd |
| Student’s t | t | df=degrees of freedom |
| Weibull | weibull | shape, scale |
Visualizing Density Function in R
The density function in R for example, dnorm() can be used to draw a graph of normal (or any distribution). Let us compare two normal distributions, both with mean = 20, one with sd = 6, and the other with sd = 3.
For this purpose, we need $x$-axis values, such as $\overline{x} \pm 3SD \Rightarrow 20 + \pm 3\times 6$.
xaxis <- seq(0, 40, 0.5) y1 <- dnorm(xaxis, 20, 6) y2 <- dnorm(xaxis, 20, 3) plot(xaxis, y2, type = "l", main = "comparing two normal distributions", col = "blue") points(xaxis, y1, type="l", col = "red")
Finding Probabilities in R
Probabilities in R language can be computed using pnorm() function for normal distribution.
#Left Tailed Probability pnorm(1.96) #Area between two Z-scores pnorm(1.96) - pnorm(-1.96)
Finding Right-Tailed Probabilities
1 - pnorm(1.96)
Solving Real Problems
Suppose you took a standardized test that has a mean of 500 and a standard deviation of 100. You got 720 marks (score). You are interested in the approximate percentile on this test.
To solve this problem, you have to find the Z-score of 720 and then use the pnorm( ) to find the percentile of your score.
zscore <- scale(x = 720, 500, 100) pnorm(2.2) pnorm(zscore[1,1]) pnorm(zscore[1]) pnorm(zscore[1, ])
Who Is This For?
✅ Data Scientists – Enhance statistical modeling
✅ Statisticians – Apply distributions in hypothesis testing
✅ R Programmers – Master distribution functions for simulations
✅ Students & Researchers – Learn with hands-on examples