R Functions for Probability Distributions
Every distribution that R handles has four functions. There is a root
name, for example, the root name for the normal distribution
is norm
. This root is prefixed by one of the letters

p
forprobability
, the distribution function (DF) 
q
forquantile
, the inverse DF for a continuous random variable or the quantile function for a discrete random variable 
d
fordensity
, the probability mass function (PMF) for a discrete random variable or the probability density function (PDF) for a continuous random variable 
r
forrandom
, a random variable having the specified distribution
For the normal distribution, these functions are
pnorm
,
qnorm
,
dnorm
, and
rnorm
.
For the binomial distribution,
these functions are
pbinom
,
qbinom
,
dbinom
, and
rbinom
.
And so forth.
For a continuous distribution (like the normal),
the most useful functions for doing problems involving probability
calculations are the
and p
functions
(DF and inverse DF), because the
the density (PDF) calculated by the
q
function can only be used to calculate probabilities
via integrals and R doesn't do integrals.
d
For a discrete distribution (like the binomial),
the
function calculates the density (PMF),
which in this case is a probability
d
and hence is useful in calculating probabilities.
R has functions to handle many probability distributions. The table below gives the names of the functions for each distribution and a link to the online documentation that is the authoritative reference for how the functions are used. But don't read the online documentation yet. First, try the examples in the sections following the table.
Distribution  Functions  

Beta  pbeta
 qbeta
 dbeta
 rbeta

Binomial  pbinom
 qbinom
 dbinom
 rbinom

Cauchy  pcauchy
 qcauchy
 dcauchy
 rcauchy

ChiSquare  pchisq
 qchisq
 dchisq
 rchisq

Discrete Uniform  sample
 
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

That's a lot of distributions. Fortunately, they all work the same way. If you learn one, you've learned them all.
Of course, the discrete distributions are discrete and the continuous distributions are continuous, so there's some difference just from that aspect alone, but as far as the computer is concerned, they're all the same. We'll do a continuous example first.
The Normal Distribtion
Direct LookUp
pnorm
is
the R function that calculates the DF.
where X is normal. Optional arguments described on the online documentation specify the parameters of the particular normal distribution.
Both of the R commands in the box below do exactly the same thing.
They look up P(X < 27.4) when X is normal with mean 50 and standard deviation 20.
Example
Question: Suppose widgit weights produced at Acme Widgit Works have weights that are normally distributed with mean 17.46 grams and variance 375.67 grams squared. What is the probability that a randomly chosen widgit weighs more then 19 grams?
Question Rephrased: What is P(X > 19) when X has the N(17.46, 375.67) distribution?
Caution: R wants the standard deviation (SD) as the parameter, not the variance. We'll need to take a square root!
Answer:
Inverse LookUp
qnorm
is
the R function that calculates the inverse DF
F^{1} of the normal distribution
The DF and the inverse DF are related by
x = F^{1}(p)
So given a number p between zero and one, qnorm
looks up the pth quantile of the normal distribution.
As with pnorm
, optional arguments specify the mean and
standard deviation of the distribution.
Example
Question: Suppose IQ scores are normally distributed with mean 100 and standard deviation 15. What is the 95th percentile of the distribution of IQ scores?
Question Rephrased: What is F^{1}(0.95) when X has the N(100, 15^{2}) distribution?
Answer:
Optional Argument lower.tail = FALSE
There is an optional argument lower.tail = FALSE
to
and p
functions. It can
be useful. It calculates the same thing as one minus the value
without this argument, but does it without
catastrophic
cancellation when the result is small.
Notice the difference.
q
Density
dnorm
is
the R function that calculates the PDF
f of the normal distribution.
As with pnorm
and qnorm
, optional arguments
specify the mean and standard deviation of the distribution.
There's not much need for this function in doing calculations, because
you need to do integrals to use any PDF, and R doesn't
do integrals. In fact, there's not much use for the
function for
any continuous distribution (discrete distributions are entirely
another matter, for them the d
functions are very useful, see
the section about dbinom).
d
For an example of the use of dnorm
, see the
following section.
Random Variates
rnorm
is
the R function that simulates random variates having a specified normal
distribution.
As with pnorm
, qnorm
, and dnorm
,
optional arguments specify the mean and standard deviation of the distribution.
We won't be using the
functions (such as r
rnorm
)
much. So here we will only give an example without full explanation.
This generates 1000 independent and identically distributed (IID) normal random numbers (first line), plots their histogram (second line), and graphs the PDF of the same normal distribution (third line).
The Binomial Distribtion
Direct LookUp, Points
dbinom
is
the R function that calculates the PMF of the binomial distribution.
Optional arguments described on the
online
documentation specify the parameters of the particular binomial
distribution.
Both of the R commands in the box below do exactly the same thing.
They look up P(X = 27) when X is has the Bin(100, 0.25) distribution.
Example
Question: Suppose widgits produced at Acme Widgit Works have probability 0.005 of being defective. Suppose widgits are shipped in cartons containing 25 widgits. What is the probability that a randomly chosen carton contains exactly one defective widgit?
Question Rephrased: What is P(X = 1) when X has the Bin(25, 0.005) distribution?
Answer:
Direct LookUp, Intervals
pbinom
is
the R function that calculates the DF of the binomial
distribution.
Optional arguments described on the
online
documentation specify the parameters of the particular binomial
distribution.
Both of the R commands in the box below do exactly the same thing.
They look up P(X ≤ 27) when X is has the Bin(100, 0.25) distribution. (Note the less than or equal to sign. It's important when working with a discrete distribution!)
Example
Question: Suppose widgits produced at Acme Widgit Works have probability 0.005 of being defective. Suppose widgits are shipped in cartons containing 25 widgits. What is the probability that a randomly chosen carton contains no more than one defective widgit?
Question Rephrased: What is P(X ≤ 1) when X has the Bin(25, 0.005) distribution?
Answer:
Optional Argument lower.tail = FALSE
See above, under continuous distributions.
Inverse LookUp
qbinom
is
the R function that calculates the quantile function
of the binomial distribution. How does it do that when the
DF is a step function and hence not invertible?
The
online
documentation for the binomial probability functions explains.
The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.
When the pth quantile is nonunique, there is a whole interval
of values each of which is a pth quantile. The documentation
says that qbinom
(and other "q
" functions,
for that matter) returns the smallest of these values. That is one
sensible definition of the quantile function.
Example
Question: What are the 10th, 20th, and so forth quantiles of the Bin(10, 1/3) distribution?
Answer:
Note the nonuniqueness.