Appendix C: Distribution Reference Card

Every statistical test assumes a distribution. This appendix is your field guide to the ones that matter in biology.

This reference covers the probability distributions you will encounter most frequently in biostatistics. For each distribution, you will find its parameters, key properties, a description of its shape, where it appears in biology, and the BioLang functions for working with it.

How to Use This Reference

Each distribution entry includes:

• Parameters — the values that define the distribution’s shape
• Mean and Variance — closed-form expressions
• Shape — what the distribution looks like
• Biological use — where this distribution appears in real data
• BioLang functions — d (density/mass), p (cumulative probability), q (quantile/inverse CDF), r (random samples)

Continuous Distributions

Normal (Gaussian)

Biological use: Gene expression levels (after log transformation), measurement errors, heights and weights in populations, many biological quantities after the central limit theorem applies.

BioLang functions:

dnorm(x, 0, 1)       # Probability density at x
pnorm(x, 0, 1)       # P(X <= x)
qnorm(p, 0, 1)       # Value at cumulative probability p
rnorm(100, 0, 1)     # Generate 100 random values

Key insight: Raw gene expression counts are not normally distributed. They follow count distributions (Poisson, negative binomial). However, log-transformed expression values (log2 CPM, log2 FPKM) are approximately normal, which is why log transformation is so common in genomics.

Log-Normal

Biological use: Fold changes in gene expression, protein concentrations, cell sizes, bacterial colony counts, drug IC50 values. Any quantity that results from multiplicative processes.

dlnorm(x, 0, 1)     # Probability density at x
plnorm(x, 0, 1)     # P(X <= x)
qlnorm(p, 0, 1)     # Value at cumulative probability p
rlnorm(100, 0, 1)   # Generate 100 random values

Key insight: When your data is right-skewed and always positive, try log-transforming it. If the log-transformed values look normal, your original data is log-normal, and you should perform statistics on the log scale.

Student’s t

Biological use: The test statistic in t-tests. Critical for small-sample inference. As df increases, the t-distribution approaches the normal distribution. With df = 30+, they are nearly identical.

dt(x, 10)            # Probability density at x
pt(x, 10)            # P(X <= x)
qt(p, 10)            # Value at cumulative probability p
rt(100, 10)          # Generate 100 random values

F Distribution

Biological use: The test statistic in ANOVA and regression F-tests. Compares the ratio of two variances. An F-value much larger than 1 indicates that between-group variance exceeds within-group variance.

df(x, 5, 20)         # Probability density at x
pf(x, 5, 20)         # P(X <= x)
qf(p, 5, 20)         # Value at cumulative probability p
rf(100, 5, 20)       # Generate 100 random values

Chi-Square

Biological use: Goodness-of-fit tests (Hardy-Weinberg equilibrium), tests of independence in contingency tables (genotype vs. phenotype associations), variance tests.

dchisq(x, 5)         # Probability density at x
pchisq(x, 5)         # P(X <= x)
qchisq(p, 5)         # Value at cumulative probability p
rchisq(100, 5)       # Generate 100 random values

Exponential

Biological use: Time between events in a Poisson process — inter-arrival times of mutations along a chromosome, time between cell divisions, radioactive decay (used in dating). The “memoryless” distribution: the probability of the event occurring in the next minute is the same regardless of how long you have been waiting.

dexp(x, 0.5)         # Probability density at x
pexp(x, 0.5)         # P(X <= x)
qexp(p, 0.5)         # Value at cumulative probability p
rexp(100, 0.5)       # Generate 100 random values

Gamma

Biological use: Waiting times for multiple events (time until k-th mutation), protein expression variance, Bayesian prior for rate parameters. Generalizes the exponential distribution (exponential is Gamma with alpha = 1).

dgamma(x, 2.0, 1.0)     # Probability density at x
pgamma(x, 2.0, 1.0)     # P(X <= x)
qgamma(p, 2.0, 1.0)     # Value at cumulative probability p
rgamma(100, 2.0, 1.0)   # Generate 100 random values

Beta

Biological use: Proportions and probabilities — allele frequencies, methylation beta-values (fraction of methylated CpGs), GC content fractions, Bayesian prior for probabilities. The natural distribution for data bounded between 0 and 1.

dbeta(x, 2.0, 5.0)      # Probability density at x
pbeta(x, 2.0, 5.0)      # P(X <= x)
qbeta(p, 2.0, 5.0)      # Value at cumulative probability p
rbeta(100, 2.0, 5.0)    # Generate 100 random values

Key insight: DNA methylation data from bisulfite sequencing produces beta-values bounded between 0 and 1. Using a beta distribution (or a beta regression) is statistically appropriate. Using a normal distribution on raw beta-values can produce nonsensical predictions outside [0, 1].

Uniform

Biological use: Null distribution for p-values (under the null hypothesis, p-values are uniformly distributed on [0, 1] — this is what Q-Q plots check), random positions along a chromosome, non-informative Bayesian priors.

dunif(x, 0, 1)       # Probability density at x
punif(x, 0, 1)       # P(X <= x)
qunif(p, 0, 1)       # Value at cumulative probability p
runif(100, 0, 1)     # Generate 100 random values

Discrete Distributions

Binomial

Biological use: Number of successes in n independent trials — number of reads mapping to a variant allele, number of patients responding to treatment, number of CpG sites methylated out of n examined.

dbinom(k, 20, 0.5)      # P(X = k)
pbinom(k, 20, 0.5)      # P(X <= k)
qbinom(q, 20, 0.5)      # Smallest k with P(X <= k) >= q
rbinom(100, 20, 0.5)    # Generate 100 random values

Poisson

Biological use: Count data when events are rare and independent — number of mutations in a genomic region, number of reads at a locus (low coverage), number of rare variants per gene, colony counts on a plate.

dpois(k, 5.0)        # P(X = k)
ppois(k, 5.0)        # P(X <= k)
qpois(q, 5.0)        # Smallest k with P(X <= k) >= q
rpois(100, 5.0)      # Generate 100 random values

Key insight: The Poisson distribution assumes that the mean equals the variance. In real RNA-seq data, the variance almost always exceeds the mean (overdispersion). This is why DESeq2 and edgeR use the negative binomial distribution instead.

Negative Binomial

Biological use: The workhorse of RNA-seq differential expression. Models count data with overdispersion (variance > mean). Used by DESeq2, edgeR, and most modern DE tools. Also models read counts in ChIP-seq, ATAC-seq, and scRNA-seq.

dnbinom(k, 10, 5)    # P(X = k), params: k, mu, size
pnbinom(k, 10, 5)    # P(X <= k)
qnbinom(q, 10, 5)    # Smallest k with P(X <= k) >= q
rnbinom(100, 10, 5)  # Generate 100 random values

Hypergeometric

Biological use: Enrichment analysis — gene ontology enrichment (Fisher’s exact test is a hypergeometric test), pathway overrepresentation, overlap between gene lists. “If I draw n genes from a genome of N, and K belong to this pathway, what is the probability of seeing k or more pathway genes by chance?”

dhyper(k, 200, 19800, 500)    # P(X = k), params: k, K, N-K, n
phyper(k, 200, 19800, 500)    # P(X <= k)
qhyper(q, 200, 19800, 500)    # Smallest k with P(X <= k) >= q
rhyper(100, 200, 19800, 500)  # Generate 100 random values

Key insight: Fisher’s exact test for 2x2 tables is equivalent to a hypergeometric test. When you run gene ontology enrichment and see a “Fisher’s exact p-value,” the software is computing hypergeometric probabilities.

Quick Reference Table

Relationships Between Distributions

Understanding how distributions relate to each other helps with intuition:

• Poisson is the limit of Binomial as n goes to infinity and p goes to 0 with n*p = lambda
• Exponential is Gamma with alpha = 1
• Chi-Square with df = k is Gamma with alpha = k/2 and beta = 1/2
• Normal is the limit of Student’s t as df goes to infinity
• Normal approximates Binomial when np > 5 and n(1-p) > 5
• Normal approximates Poisson when lambda > 20
• Negative Binomial reduces to Poisson when size goes to infinity (no overdispersion)
• Hypergeometric approaches Binomial when N is much larger than n

# Visualize the Poisson-Normal convergence
[1, 5, 10, 30] |> each(|lam| {
  let data = rpois(10000, lam)
  histogram(data, {title: "Poisson(lambda=" + str(lam) + ")", bins: 30})
})