Mastering Probability Distributions in R: From Normal to Poisson

Random variable measurements follow specific distributions; many continuous variables in life follow the normal distribution, such as human height or crop yield.

Discrete variables follow distributions like binomial or Poisson; for example, gene expression counts from RNA‑seq can be approximated by a Poisson distribution, while differential expression analysis often uses a negative binomial model.

Statistics includes many distributions such as normal, binomial, Poisson, chi‑square, etc. Knowing a distribution allows calculation of probability density, cumulative distribution function (CDF), quantiles, and generation of random samples; R provides corresponding functions. In R, function names consist of a prefix and a suffix: the prefix d, p, q, r indicates density, CDF, quantile, and random generation respectively, and the suffix denotes the distribution type (e.g., dnorm for normal).

Common distributions and their R functions include:

Normal (norm): dnorm, pnorm, qnorm, rnorm

Binomial (binom): dbinom, pbinom, qbinom, rbinom

Poisson (pois): dpois, ppois, qpois, rpois

Hypergeometric (hyper): dhyper, phyper, qhyper, rhyper

Chi‑square (chisq): dchisq, pchisq, qchisq, rchisq

t distribution (t): dt, pt, qt, rt

F distribution (f): df, pf, qf, rf

Negative binomial (nbinom): dnbinom, pnbinom, qnbinom, rnbinom

Exponential (exp): dexp, pexp, qexp, rexp

Example: A random variable follows a standard normal distribution; compute the quantile for a CDF of 0.95.

<code>qnorm(0.95)
# 1.64485362695147</code>

The syntax of qnorm() is qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE) . In the example the default parameters mean = 0, sd = 1, lower.tail = TRUE are used. The parameter mean denotes the distribution mean, sd the standard deviation, and lower.tail indicates whether to compute the lower‑tail quantile.

Example: In a region with one million people, about 40 per 100,000 are diagnosed with lung cancer each year. Assuming new cases follow a Poisson distribution with λ = 40, what is the probability that fewer than 200 cases occur in a year?

<code>ppois(20, 40)</code>

The syntax of ppois() is ppois(q, lambda, lower.tail = TRUE, log.p = FALSE) . Here λ = 40 represents the expected number of new lung‑cancer diagnoses per 100,000 people per year; using the Poisson CDF yields the probability that annual new cases are below 200.

Source: Liu Hongde, Sun Xiao, Xie Jianming, “Bioinformatics Data Analysis and Practice”.