Probability Distributions in R

R has a consistent interface for probability distributions. The book uses probability as a bridge between programming and statistics: once you can create vectors, call functions, and plot results, you can compute probabilities, quantiles, simulated samples, and distribution curves. This makes abstract probability concrete because every distribution family has a small set of related functions.

Most distribution functions come in four forms. For a distribution root such as norm, R provides dnorm for density, pnorm for cumulative probability, qnorm for quantiles, and rnorm for random generation. The same prefix pattern appears for binomial (binom), Poisson (pois), uniform (unif), exponential (exp), t (t), chi-square (chisq), F (f), and many others.

Definitions

A probability distribution describes possible values of a random variable and how probability is assigned to those values.

A discrete distribution assigns probability to countable values. Examples include Bernoulli, binomial, and Poisson distributions. R's d functions return probabilities for discrete distributions.

A continuous distribution assigns probability over intervals. Examples include normal, uniform, exponential, and t distributions. R's d functions return density values, not interval probabilities.

A cumulative distribution function (CDF) returns $P(X \le x)$ . In R, p* functions compute CDF values, such as pnorm(1.96).

A quantile function is the inverse CDF. It returns the value $x$ such that $P(X \le x) = p$ . In R, q* functions compute quantiles, such as qnorm(0.975).

A random sample generator creates pseudo-random observations from a distribution. In R, r* functions do this, such as rbinom(100, size = 10, prob = 0.3).

Key results

R's distribution naming pattern is the main result:

Prefix	Meaning	Normal example	Binomial example
`d`	density or probability mass	`dnorm(x, mean, sd)`	`dbinom(x, size, prob)`
`p`	cumulative probability	`pnorm(q, mean, sd)`	`pbinom(q, size, prob)`
`q`	quantile	`qnorm(p, mean, sd)`	`qbinom(p, size, prob)`
`r`	random generation	`rnorm(n, mean, sd)`	`rbinom(n, size, prob)`

For continuous distributions, point probabilities are zero. dnorm(0) is a density height, not $P(X = 0)$ . To get interval probabilities, subtract CDF values:

\begin{aligned} P(a < X < b) &= F(b) - F(a). \end{aligned}

For discrete distributions, dbinom(3, size = 10, prob = 0.4) is the probability of exactly 3 successes. A cumulative statement such as "at most 3 successes" uses pbinom(3, size = 10, prob = 0.4).

Simulation should be reproducible when used in notes, reports, or tests. set.seed() fixes the pseudo-random sequence so readers can reproduce the same sample.

Visual

Distribution	R root	Parameters	Common use
Normal	`norm`	`mean`, `sd`	Measurement variation, approximations
Binomial	`binom`	`size`, `prob`	Number of successes in fixed trials
Poisson	`pois`	`lambda`	Event counts per interval
Uniform	`unif`	`min`, `max`	Equal chance over interval
Exponential	`exp`	`rate`	Waiting times
t	`t`	`df`	Inference for means with estimated variance

Worked example 1: Normal interval probability

Problem: suppose exam scores are approximately normal with mean 75 and standard deviation 8. Find the probability that a randomly chosen score is between 70 and 90.

Method:

Define $X \sim N(75, 8^2)$ .
Use the CDF identity $P(70 \lt X \lt 90) = F(90) - F(70)$ .
Compute both cumulative probabilities with pnorm.
Subtract.
Sanity-check using z-scores.

lower <- pnorm(70, mean = 75, sd = 8)
upper <- pnorm(90, mean = 75, sd = 8)
prob <- upper - lower

lower
# [1] 0.2659855

upper
# [1] 0.9696036

prob
# [1] 0.7036181

Checked answer: the lower z-score is (70 - 75) / 8 = -0.625, and the upper z-score is (90 - 75) / 8 = 1.875. The interval spans from moderately below the mean to well above it, so a probability around 0.70 is plausible. The computed answer is approximately 0.704.

The key is that dnorm(70) and dnorm(90) would not answer this question; they are density heights. Interval probability comes from pnorm.

Worked example 2: Binomial exact and cumulative probabilities

Problem: a machine produces an acceptable part with probability 0.9. In 12 independent parts, find the probability of exactly 10 acceptable parts and the probability of at least 10 acceptable parts.

Method:

Define $X \sim \mathrm{Binomial}(12, 0.9)$ .
Use dbinom for exactly 10.
Use either a sum of dbinom(10:12, ...) or a CDF complement for at least 10.
Check that both cumulative methods agree.

exact_10 <- dbinom(10, size = 12, prob = 0.9)
at_least_10_sum <- sum(dbinom(10:12, size = 12, prob = 0.9))
at_least_10_cdf <- 1 - pbinom(9, size = 12, prob = 0.9)

exact_10
# [1] 0.2301278

at_least_10_sum
# [1] 0.8891308

at_least_10_cdf
# [1] 0.8891308

Checked answer: "at least 10" means 10, 11, or 12 acceptable parts. Summing those exact probabilities matches the complement of "at most 9." The result is high, about 0.889, because the success probability is 0.9 and 12 trials is not large.

This example also shows the value of naming variables. exact_10 and at_least_10_cdf prevent confusion between point and cumulative probabilities.

Code

# Compare theoretical and simulated binomial probabilities.

set.seed(123)
n_sim <- 10000
size <- 12
prob <- 0.9

simulated <- rbinom(n_sim, size = size, prob = prob)
sim_table <- prop.table(table(factor(simulated, levels = 0:size)))
theory <- dbinom(0:size, size = size, prob = prob)

comparison <- data.frame(
  successes = 0:size,
  simulated = as.numeric(sim_table),
  theoretical = theory
)

print(tail(comparison, 5))

plot(
  comparison$successes,
  comparison$theoretical,
  type = "h",
  lwd = 4,
  xlab = "Acceptable parts",
  ylab = "Probability",
  main = "Binomial(12, 0.9) probabilities"
)
points(comparison$successes, comparison$simulated, pch = 19)

The simulation table is a useful bridge between probability theory and data analysis. The theoretical column is what the binomial model says before data are simulated. The simulated column is what happened in 10,000 pseudo-random experiments. The two columns should be close, but not identical, because simulation has sampling variability. Increasing n_sim usually makes the simulated proportions move closer to the theoretical probabilities.

This code also illustrates why factor levels are supplied in table(factor(simulated, levels = 0:size)). Without explicit levels, outcomes that did not occur in the simulation would be absent from the table, shortening the result and misaligning it with 0:size. By declaring all possible success counts, the simulated and theoretical vectors have the same length and refer to the same outcomes.

When using distribution functions, name the event in words before coding it. "Exactly 3" becomes dbinom(3, ...). "At most 3" becomes pbinom(3, ...). "More than 3" becomes pbinom(3, ..., lower.tail = FALSE) or 1 - pbinom(3, ...). "Between 70 and 90" for a continuous variable becomes a difference of two CDF values. Translating the event first prevents off-by-one errors and wrong-tail errors.

Simulation is also a diagnostic tool. If a theoretical probability calculation seems suspicious, simulate the experiment and compare the approximate frequency. Disagreement usually means either the formula was coded incorrectly or the simulation does not match the same assumptions.

Distribution functions also support inference pages. Critical values in confidence intervals come from quantile functions such as qt and qnorm. P-values often come from cumulative probabilities such as pt, pnorm, or pchisq. Random generators support power simulations and teaching demonstrations. Learning the d, p, q, r pattern once pays off across most of introductory statistics.

Parameterization must be checked from the help page. For example, normal functions use mean and sd, while exponential functions use rate by default, and some distributions offer alternative parameter names. A mathematically correct idea can become wrong code if the parameter is supplied on the wrong scale. When in doubt, run a small sanity check, such as verifying the simulated mean is close to the theoretical mean.

Graphing a distribution is often the fastest sanity check. A discrete mass function should have bars at possible values whose heights sum to one. A continuous density should have plausible shape and scale, but areas, not point heights, represent probabilities. Pairing probability calculations with plots makes tail direction and endpoint choices easier to verify.

When a result is a probability, check that it lies between 0 and 1. When a result is a quantile, check that it is on the original measurement scale. This simple distinction catches many prefix mix-ups.

Common pitfalls

Treating a continuous density value as a probability. Use CDF differences for intervals.
Confusing lower.tail = FALSE with subtracting the wrong endpoint. For P(X > q), use lower.tail = FALSE or 1 - p*(q).
Forgetting distribution parameters. sd in rnorm is standard deviation, not variance.
Using set.seed() once and then expecting every later random call to be independent of earlier calls in the script.
Simulating a small number of observations and expecting exact agreement with theoretical probabilities.
Using pbinom(10, ...) for "at least 10"; that gives "at most 10."

Definitions​

Key results​

Visual​

Worked example 1: Normal interval probability​

Worked example 2: Binomial exact and cumulative probabilities​

Code​

Common pitfalls​

Connections​