How to Calculate Probabilities for Continuous Distributions in R


To calculate probabilities for continuous distributions, you need to use the cumulative distribution function, CDF. Unlike discrete distributions, there is no single point probability, P(X=x), there is a density. In this article, we will learn how to calculate probabilities of continuous distributions in R.


To calculate these probabilities, we can use function prefixed with p for each distribution. For example, to calculate the cdf of the normal distribution, we can use pnorm.

pnorm(q = .8, mean = 0, sd = 1)
## [1] 0.7881446

This calculates the probability of selecting an item below .8 in the normal 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 qninom.

qnorm(0.7881446, mean = 0, sd = 1)
## [1] 0.8

Notice that our result is .8 which is exactly the value of q from the previous section. This shows how the two functions are inversely related.

Generating Random Samples

It is common to generate a random sample of normal variables. These are used in simulations and many other situations you will learn. To do this, we can follow a similar pattern of the above function using the norm preceded by a r for random.

##  [1]  0.86408802 -0.50945900  0.84883333  0.37336024  1.43838537  1.11448386
##  [7]  0.46681953  0.75352516  0.25414154  0.09360259

Here we generated sample of size 10 from the normal distributions.


The pdf isn’t very useful for continuous variables as you are usually looking for the probability which is the cumulative density. However, you can use the dnorm and other functions to generate the density plots.

x <- seq(-4, 4, length=100)
y <- dnorm(x)
plot(y, type='l')

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