Random Variables and Probability Distributions

A random variable is a function that assigns a numerical value to every possible outcome of a chance process. It is not the outcome itself — "heads" is not a number — but a rule for turning outcomes into numbers. Conventionally written with a capital letter ($X$, $Y$), a random variable becomes a specific realized number (lowercase, $x$) once the experiment happens. If you count the number of albino offspring in a litter of 6 mice from a heterozygous cross, $X$ is "number of albino pups," and it can realize as $x = 0, 1, 2, \dots, 6$. Random variables split into two families based on what values they can take. A discrete random variable takes on a countable set of values — often (but not always) non-negative integers: number of mutations in a genome, number of surviving seedlings, number of colonies on a plate. A continuous random variable can take any value in an interval of the real line — height, enzyme velocity, time to death, blood glucose concentration. The distinction matters enormously for probability: for a discrete variable, individual values can have positive probability, but for a continuous variable, any single exact value has probability zero — probability only accumulates over intervals. This single fact explains why continuous distributions are described by density curves rather than lists of probabilities. The complete description of a random variable's behavior is its probability distribution — the full mapping from values (or intervals) to probabilities. For discrete variables this is the probability mass function (pmf), $p(x) = P(X = x)$, satisfying $p(x) \ge 0$ for all $x$ and $\sum_x p(x) = 1$. For continuous variables it is the probability density function (pdf), $f(x) \ge 0$, satisfying $\int_{-\infty}^{\infty} f(x)\,dx = 1$, where probability corresponds to area under $f(x)$ over an interval, $P(a \le X \le b) = \int_a^b f(x)\,dx$. Both types also have a cumulative distribution function (CDF), $F(x) = P(X \le x)$, which is non-decreasing, starts at 0, and approaches 1 — the CDF is the single most useful tool for reading off probabilities of ranges directly, whether the underlying variable is discrete or continuous.

Step 1: Define the sample space and the random variable's rule. Start from the underlying chance process (a cross, a count in a volume of blood, a measurement of an organism) and specify exactly which number $X$ reports. Precision matters: "number of red-eyed flies in a vial of 40" is a well-defined discrete random variable; "eye color" alone is not numerical and is not a random variable until you code it (e.g., $X=1$ for red, $X=0$ for white).

Step 2: Assign probabilities consistently with the two axioms. For a discrete variable, every possible value $x_i$ gets a probability $p(x_i)$, and these must sum to 1 across the whole support (the set of possible values). For a continuous variable, a smooth density $f(x)$ is proposed (often derived from a mechanistic model — a genetic ratio, a rate process, a limiting Gaussian) and normalized so total area is 1. This normalization step is not optional: any function used as a pmf or pdf must be checked or scaled so probabilities are internally consistent.

Step 3: Compute the expected value — the probability-weighted center. For discrete $X$:

For continuous $X$:

$E[X]$ is often called $\mu$. It is not necessarily a value $X$ can actually take — the expected number of albino pups from a 3:1 cross of litter size 4 is $1.0$, achievable, but the expected number of heads in 3 coin flips is $1.5$, which a single litter can never show. $E[X]$ describes the long-run average over many repetitions, consistent with the frequentist interpretation of probability introduced alongside conditional probability.

Step 4: Compute the variance — the probability-weighted spread around the mean.

A computationally convenient identity, used constantly in practice, is:

The square root of the variance is the standard deviation, $\sigma = \sqrt{\mathrm{Var}(X)}$, expressed in the original units of $X$ (variance itself is in squared units, which is why it is harder to interpret directly — a variance of "4 pups²" is awkward, but $\sigma = 2$ pups is intuitive).

Step 5: Use the CDF to answer range questions. Biological questions rarely ask "what is the probability of exactly this value" for continuous data — they ask "what fraction of patients have LDL above 160?" or "what is the probability of 3 or fewer mutant colonies?" The CDF answers this directly:

This is the operational bridge to named distributions: once biologists recognize a process fits a binomial, Poisson, or normal pattern, they use tabulated or software-computed CDFs rather than deriving sums or integrals from scratch every time.

Step 6: Recognize linear transformations and their effect on mean and variance. If $Y = aX + b$ (e.g., converting a cell count $X$ to a concentration $Y$ by scaling and shifting):

Note variance scales by $a^2$, not $a$ — doubling a measurement scale quadruples the variance, a fact that matters when biologists change units (e.g., mg/dL to mmol/L) and need to recompute spread correctly.

Unlike a single landmark wet-lab experiment (Meselson–Stahl, Hershey–Chase), the theory of random variables and probability distributions was validated through a convergence of mathematical proof and a century of applied biological testing, most visibly in genetics. Three episodes show the framework earning its keep.

Gregor Mendel's pea crosses (1856–1863) as the first large-scale test of a discrete probability model in biology. Mendel crossed true-breeding pea plants differing in traits like seed shape and color, then counted offspring phenotypes across thousands of plants. For a monohybrid cross ($Aa \times Aa$), Mendel predicted a discrete random variable — number of recessive offspring among $n$ trials — should follow ratios of 3:1 for dominant:recessive phenotypes. In one of his largest experiments, counting seed shape in 7,324 seeds from self-fertilized hybrids, he observed 5,474 round and 1,850 wrinkled — a ratio of 2.96 : 1, extremely close to the predicted 3:1. This was among the earliest demonstrations that a biological outcome could be modeled as a random variable with a known pmf (what would later be formalized as the binomial distribution), and that large samples converge toward theoretical probabilities — an empirical confirmation of the frequentist idea that relative frequencies approach true probabilities as $n$ grows.

R.A. Fisher's 1936 statistical re-analysis of Mendel's data. Fisher computed a chi-square goodness-of-fit statistic across Mendel's pooled experiments and found the aggregate fit to expected ratios was suspiciously good — closer to the theoretical 3:1 (and other predicted ratios) than random sampling variance should typically allow. This became known as the "too good to be true" problem. Fisher's analysis is itself powerful evidence for the framework: it shows that once a probability distribution is specified (the binomial pmf implied by Mendelian ratios), deviations from it are quantifiable, and even a goodness of fit that is too perfect is statistically detectable — proof that variance is a real, expected feature of any correctly specified random process, not an artifact to be minimized away. This episode is frequently revisited in biostatistics as a caution: an absence of expected scatter is itself informative, illustrating that $\mathrm{Var}(X) > 0$ is a prediction, not a nuisance.

Adolphe Quetelet and Francis Galton's demonstration that continuous biological measurements follow smooth density curves (1840s–1880s). Quetelet measured chest circumferences of 5,738 Scottish soldiers and showed the frequency histogram closely tracked a bell-shaped curve — the earliest clear empirical case for treating a continuous biological trait as a random variable governed by a probability density rather than a fixed value. Galton extended this to human height and to sweet-pea seed diameters, showing that continuous variation in a biological trait clusters symmetrically around a mean with a characteristic spread — a phenomenon later formalized as the normal distribution. The critical empirical point these studies established, and which underlies this whole chapter, is that variability itself is lawful: biological measurement scatter is not noise to be dismissed, but obeys a describable, quantifiable, and predictable mathematical density. Together, these three lines of evidence show the two branches of the framework validated independently: discrete counting processes (Mendel) match pmf predictions, deviations from those predictions are themselves statistically tractable (Fisher), and continuous biological measurements match smooth density curves (Quetelet, Galton) — exactly the discrete/continuous split this article opened with.

Example 1 — A genetics cross with an asymmetric, non-Mendelian twist (discrete random variable). A geneticist studies a gene in mice where the heterozygote is not a simple dominant/recessive case but shows incomplete penetrance: individuals with genotype $Aa$ display the mutant phenotype with probability 0.6 (not 1.0), while $aa$ individuals always show it and $AA$ individuals never do. A cross $Aa \times Aa$ gives genotypes in the classic ratio 1 $AA$ : 2 $Aa$ : 1 $aa$. Let $X = 1$ if a randomly chosen offspring shows the mutant phenotype, $X = 0$ otherwise. Compute the pmf of $X$ by conditioning on genotype (a direct application of the law of total probability, extending Conditional Probability):

So $P(X=1) = 0.55$, $P(X=0) = 0.45$. This is a Bernoulli random variable with $p = 0.55$ — not the naive 0.75 a student might guess from "3/4 show the phenotype," because incomplete penetrance shrinks the effective probability.

So across many offspring, 55% are expected to show the phenotype, with variance 0.2475 per individual — this single-offspring Bernoulli variable is exactly the building block that, summed over $n$ independent offspring, produces the Binomial Distribution.

Example 2 — A continuous physiological variable and a clinical threshold (continuous random variable). Suppose resting heart rate $X$ (beats per minute) in a population of adult patients is modeled with the density

(a uniform distribution — a simplification, since real heart rate is closer to normal, but it is analytically transparent and satisfies both pdf axioms: $f(x) \ge 0$, and $\int_{50}^{70}\frac{1}{20}\,dx = \frac{20}{20}=1$.)

Expected value:

(For a uniform distribution this matches the simple midpoint, $(50+70)/2 = 60$, as expected.)

Variance, using $\mathrm{Var}(X) = \dfrac{(b-a)^2}{12}$ for a uniform variable on $[a,b]$:

Clinical question: what is the probability a randomly selected patient has a resting heart rate above 65 bpm (a common "watch" threshold)?

So 25% of this patient population is expected to have resting heart rate above 65 bpm — computed directly from the density without needing a table, because for a uniform distribution probability is just proportional interval length. This same logic — probability as an area, computed via the CDF, $F(70)-F(65)$ — generalizes directly to the Normal Distribution, where the density is not flat but bell-shaped and the areas must be read from a standardized table or computed numerically.

The modern axiomatic treatment of random variables and probability distributions was formalized by Andrey Nikolaevich Kolmogorov, the Soviet mathematician who in 1933 published Grundbegriffe der Wahrscheinlichkeitsrechnung (Foundations of the Theory of Probability), placing probability on a rigorous measure-theoretic footing for the first time. Before Kolmogorov, probability calculations in biology (Mendel's ratios, Quetelet's and Galton's curves) worked empirically and intuitively but lacked a unifying axiomatic base; after Kolmogorov, the pmf/pdf/CDF machinery used throughout biostatistics could be derived from three simple axioms (non-negativity, total probability 1, and additivity over disjoint events) rather than assumed piecemeal for each new problem. Kolmogorov was not a biologist, but his framework became the mathematical substrate on which R.A. Fisher, Sewall Wright, and J.B.S. Haldane — the founders of population genetics and biometrical genetics — built quantitative models connecting Mendelian inheritance to continuous trait variation, unifying the discrete world of gene counting and the continuous world of measured phenotypes into the single theory presented in this article.

Clinical diagnostics and reference ranges. Laboratory test results (blood glucose, cholesterol, hemoglobin) are treated as continuous random variables with population-specific densities; "normal range" cutoffs are literally computed as CDF percentiles (commonly the 2.5th and 97.5th) of a fitted distribution, directly using the $F(x)=P(X\le x)$ machinery in this article.

Genetic counseling and carrier-risk calculation. Predicting the probability a couple's child will inherit a recessive disorder uses discrete random variables built from pedigree-conditioned probabilities, extending the Bernoulli-and-binomial logic of the worked cross example to real clinical risk communication.

Quality control in biomanufacturing. Counts of contaminant colonies on a plate, defective cells in a bioreactor batch, or viral particles in a vaccine lot are modeled as discrete random variables (often Poisson-like, foreshadowed here), with acceptance thresholds set from the pmf/CDF.

Ecology and conservation. Species abundance counts in quadrat sampling, number of nesting pairs observed, or mutation counts across a genome are discrete random variables whose expected value and variance drive population viability estimates and extinction-risk models.

Epidemiology and public health. Prevalence and incidence calculations, vaccine efficacy trials, and dose-response curves for toxicology all rest on correctly distinguishing discrete case-counts from continuous exposure measurements, and on computing means, variances, and probabilities from properly normalized distributions — the exact toolkit this article establishes before moving to the binomial, Poisson, and normal distributions used to name specific real-world processes.

:::quiz[A Bernoulli random variable $X$ has $P(X=1)=0.55$ (mutant phenotype shown) due to incomplete penetrance. What is $E[X]$?]

• 1.0
• 0.6
• 0.55 ✓
• 0.3025

Ask the tutor: How do I decide, for a new biological measurement, whether to model it as discrete or continuous — and does that choice change how I compute risk thresholds?