Probability Distributions in R

R provides a systematic way to work with probability distributions.
For each distribution, there are four types of functions, following a naming convention:

1. Density / Probability Mass (dxxx)

  • For continuous distributions (like the normal), this is the probability density function (pdf).
    It does not give a probability directly, but tells you how “dense” the distribution is around a value.
    Example: dnorm(0) gives the height of the normal curve at 0.
  • For discrete distributions (like the binomial or poisson), it is the probability mass function (pmf).
    This gives the actual probability of the outcome.
    Example: dbinom(3, size=10, prob=0.5) = probability of exactly 3 successes.

Think of dxxx as answering: “What is the likelihood at this exact point?”

# Normal density at x = 0, mean = 0, sd = 1
dnorm(0, mean = 0, sd = 1)
[1] 0.3989423
# Probability that a Poisson(λ=2) equals x = 3
dpois(3, lambda = 2)
[1] 0.180447

2. Cumulative probability (pxxx)

  • This function gives the probability that the random variable is less than or equal to a value.

  • Example: pnorm(1.96) = probability that a standard normal random variable is ≤ 1.96 (about 0.975).

  • Example: pbinom(4, size=10, prob=0.5) = probability of getting up to 4 successes in 10 coin tosses

Think of pxxx as answering:
“What is the probability that I am at or below this value?”

# P(X ≤ 1.96) when X ~ N(0,1)
pnorm(1.96, mean = 0, sd = 1)
[1] 0.9750021
# Cumulative probability for Binomial(n=10, p=0.3), X ≤ 4
pbinom(4, size = 10, prob = 0.3)
[1] 0.8497317

3. Quantiles (qxxx)

  • This is the inverse of the CDF.

  • You give it a probability p, and it tells you the corresponding value (quantile).

  • Example: qnorm(0.975) = the 97.5th percentile of the normal distribution (≈ 1.96).

  • Example: qbinom(0.5, size=10, prob=0.5) = the median number of successes in 10 coin tosses.

Think of qxxx as answering:
“At what value do I reach this probability?”

# 97.5th percentile of a N(0,1)
qnorm(0.975, mean = 0, sd = 1)
[1] 1.959964
# Median of Chi-squared with 5 df
qchisq(0.5, df = 5)
[1] 4.35146

4. Random generation (rxxx)

  • This generates random numbers following the distribution.

  • Example: rnorm(5, mean=0, sd=1) → five random draws from a standard normal distribution.

  • Example: rpois(10, lambda=3) → ten simulated counts from a Poisson distribution.

Think of rxxx as answering:
“Give me some random data from this distribution.”

# Generate 5 random normal values
rnorm(5, mean = 0, sd = 1)
[1]  0.482202123  1.140005813  1.158857672  0.003924625 -1.026943908
# Generate 10 random uniform values between 0 and 1
runif(10, min = 0, max = 1)
 [1] 0.82426720 0.52138493 0.91966684 0.84333101 0.47440892 0.80970328
 [7] 0.07018974 0.05150087 0.63673564 0.71285348

Common Distributions and Function Prefixes

Distribution Prefix Example Functions
Normal norm dnorm, pnorm, qnorm, rnorm
Binomial binom dbinom, pbinom, qbinom, rbinom
Poisson pois dpois, ppois, qpois, rpois
Exponential exp dexp, pexp, qexp, rexp
Uniform unif dunif, punif, qunif, runif
Chi-squared chisq dchisq, pchisq, qchisq, rchisq
t-distribution t dt, pt, qt, rt
F-distribution f df, pf, qf, rf
Gamma gamma dgamma, pgamma, qgamma, rgamma
Beta beta dbeta, pbeta, qbeta, rbeta
Geometric geom dgeom, pgeom, qgeom, rgeom
Hypergeometric hyper dhyper, phyper, qhyper, rhyper

Right now, these distributions may seem very theoretical. In chapter P-values and Confidence Intervals however, we’ll use these to get some actual inference from the STEpS data.