Discrete Distributions in R


Two common probabilities you will calculate in statistics is P(X = x) and P(X ≤ x). R provides your some simple ways to calculate these probabilities. In this article, we will learn how to calculate probabilities for Discrete Distributions in R.


To calculate P(X = x), we can use the density functions. R has many functions for this all prefixed with d. For example, we can use dbinom to calculate the binomial distribution.

dbinom(7, size = 100, prob = 0.5)
## [1] 1.262774e-20

The above shows us the P(X = 7) when X is from a Bin(100, .5) distribution.

The CDF or Probability Mass Function

To calculate the probability that an item is less than x or P(X ≤ x), we can use the probability mass function, or the CDF. R has many methods all prefixed with p. For example, we can use pbiom for the binomial distribution.

pbinom(7, size = 10, prob = 0.5)
## [1] 0.9453125

The above shows us the P(X ≤ 7) when X is from a Bin(10, .5) distribution.

The Quantile or the Inverse CDF

Next, we can take a look at getting the quantile from a distribution. This is also known as the inverse CDF. To do this, we can use the distribution functions prefixed with a q. In our case, we will use qbinom.

qbinom(0.9453125, size = 10, prob = 0.5)
## [1] 7

In the above example, we put in the P(X ≤ 7) which we got from our previous example. We can see the value returned is 7 which is the value of X we used in our pdf.

Generating Random Binomials

We can also generate random samples from any of the distributions. Again, we follow the same pattern of having a short name for the distribution, i.e. binom, preceded by a prefix, r. To generate random samples, we can use rbinom.

rbinom(10, size =  10, prob = .5)
##  [1] 2 3 8 6 9 6 4 6 4 7

The example above will generate a random sample, which is multiple values of X from a Bin(10, .5) distribution. In our case, we supplied 10 as the first argument, thus our sample size will be 10.

Other Distributions

R provides many other continuous distributions with the same structure as above. Here is a list of many distributions and their respective functions.

Distribution Functions
Beta pbeta qbeta dbeta rbeta
Binomial pbinom qbinom dbinom rbinom
Cauchy pcauchy qcauchy dcauchy rcauchy
Chi-Square pchisq qchisq dchisq rchisq
Exponential pexp qexp dexp rexp
F pf qf df rf
Gamma pgamma qgamma dgamma rgamma
Geometric pgeom qgeom dgeom rgeom
Hypergeometric phyper qhyper dhyper rhyper
Logistic plogis qlogis dlogis rlogis
Log Normal plnorm qlnorm dlnorm rlnorm
Negative Binomial pnbinom qnbinom dnbinom rnbinom
Normal pnorm qnorm dnorm rnorm
Poisson ppois qpois dpois rpois
Student t pt qt dt rt
Studentized Range ptukey qtukey dtukey rtukey
Uniform punif qunif dunif runif
Weibull pweibull qweibull dweibull rweibull
Wilcoxon Rank Sum Statistic pwilcox qwilcox dwilcox rwilcox
Wilcoxon Signed Rank Statistic psignrank qsignrank dsignrank rsignrank