Agricultural Statistics & Data Science · Statistical theory core · also Distribution Theory, Statistical Methods I (part)
Every standard distribution a statistician must recognise and derive from: discrete and continuous families, their relations and limiting forms, compound and truncated distributions, order statistics, the bivariate normal and the sampling distributions chi-square, t and F.
In the exams: AICE-JRF Ag Stat Unit 1 and Unit 2 'compound and truncated distribution, order statistics' (heavy); GATE ST distributions and order statistics (heavy); IIT JAM MS (heavy); UPSC ISS Statistics I (heavy)
Distribution theory is the catalogue and the grammar of statistical models. The catalogue is the set of standard families (binomial, Poisson, negative binomial, normal, gamma, beta and the rest) with their moments, generating functions and shapes. The grammar is the set of relations that connect them: limits, sums, ratios, mixtures, truncations and ordered values. A statistician who knows both can look at a count of aphids, a set of plot yields or a residual sum of squares and say at once what its distribution is and why.
Agriculture supplies the examples. Germination of seeds in a tray is binomial; insects per leaf are Poisson when they are scattered at random and negative binomial when they clump; times to failure of irrigation pumps are exponential or Weibull; percentages of infested plants follow beta-type shapes; the largest daily rainfall of a season is an order statistic; and the sums of squares of every analysis of variance in a field-trial report are chi-square variables whose ratios are F, by Cochran's theorem.
This manual moves from the standard discrete and continuous families to their relations and limiting forms, compound, mixture and truncated distributions, the Pearson system and the fitting of distributions to agricultural data, order statistics, the bivariate normal, the sampling distributions chi-square, t and F with their non-central versions, the exponential family and, finally, the distribution of quadratic forms in normal variables. Probability tools (generating functions, transformations, limit theorems) are developed in the companion subject Probability Theory and are used here freely.
Examinations everywhere test this material heavily. GATE Statistics, IIT JAM Mathematical Statistics, the ISI tests and the Chinese postgraduate statistics paper ask for moments, MGF identification, transformation results and order-statistic densities as numerical and multiple-select items. The ICAR AICE-JRF/SRF Agricultural Statistics paper names distributions in Unit 1 and 'compound and truncated distributions, order statistics' in Unit 2. The Indian Statistical Service, state statistical services and the ICAR entrance tests ask for properties of binomial, Poisson and normal distributions and for fitting them to data.
A Bernoulli trial has two outcomes, success with probability p and failure with probability q = 1 − p; the indicator X of success has mean p and variance pq. The binomial distribution counts successes in n independent Bernoulli trials with constant p: P(X = x) = C(n, x)pˣqⁿ⁻ˣ, x = 0, 1, ..., n. Its mean is np, its variance npq (always less than the mean), its MGF (q + peᵗ)ⁿ, μ₃ = npq(q − p), β₁ = (q − p)²/(npq) and β₂ = 3 + (1 − 6pq)/(npq). It is symmetric when p = 1/2, positively skewed when p < 1/2. Successive probabilities obey the recurrence P(x + 1)/P(x) = [(n − x)/(x + 1)](p/q), which is how expected frequencies are computed when fitting. The mode is the integer part of (n + 1)p; when (n + 1)p is an integer there are two modes, (n + 1)p and (n + 1)p − 1. Independent binomials with the same p add: Bin(n₁, p) + Bin(n₂, p) = Bin(n₁ + n₂, p).
The binomial needs a fixed n, independence and constant p. Seeds in one germination tray satisfy these reasonably; plants in a row affected by a spreading disease do not, because infection of one raises the chance for its neighbours.
The multinomial generalises the binomial to k categories with probabilities p₁, ..., p_k: P(x₁, ..., x_k) = n!/(x₁!⋯x_k!) p₁^x₁⋯p_k^x_k with Σxᵢ = n. Each marginal Xᵢ is binomial(n, pᵢ), Var(Xᵢ) = npᵢ(1 − pᵢ) and Cov(Xᵢ, Xⱼ) = −npᵢpⱼ: the counts are negatively correlated because they share a fixed total. The progeny of a dihybrid cross classified into four phenotypes with expected ratio 9:3:3:1 is the standard genetics example.
The Poisson distribution, P(X = x) = e^(−λ)λˣ/x!, x = 0, 1, 2, ..., describes counts of events occurring at random in time or space at a constant rate: insects per leaf when they are scattered at random, weed seedlings per quadrat, defects per metre of drip line, calls to a veterinary helpline per hour. Its mean and variance are both λ; μ₃ = λ, β₁ = 1/λ and β₂ = 3 + 1/λ, so it is positively skewed and approaches symmetry as λ grows. The MGF is exp(λ(eᵗ − 1)). The recurrence P(x + 1) = λP(x)/(x + 1) gives fitted frequencies quickly. The mode is the integer part of λ, with two modes λ and λ − 1 when λ is an integer. Independent Poisson variables add: their sum is Poisson with the summed mean.
Equality of mean and variance is the Poisson signature. The ratio s²/x̄, the index of dispersion, is near 1 for random (Poisson) patterns, above 1 for clumped (aggregated) patterns and below 1 for regular patterns. Most insect and disease counts in the field are aggregated, which is why the negative binomial is often preferred. If X and Y are independent Poisson with means λ₁ and λ₂, the conditional distribution of X given X + Y = n is binomial(n, λ₁/(λ₁ + λ₂)).
The geometric distribution counts failures before the first success: P(X = x) = qˣp, x = 0, 1, 2, ..., with mean q/p and variance q/p². Some texts count trials up to and including the first success (support 1, 2, ...), giving mean 1/p and the same variance; always check which version a question uses. The geometric is the only discrete distribution with the memoryless property: P(X ≥ s + t | X ≥ s) = P(X ≥ t).
The negative binomial counts failures before the r-th success: P(X = x) = C(x + r − 1, x)pʳqˣ, x = 0, 1, 2, ..., with mean rq/p and variance rq/p², which exceeds the mean. It is the sum of r independent geometric variables. In ecology it is parameterised by its mean m and an aggregation parameter k (playing the role of r, but allowed to be any positive real): variance m + m²/k. Small k means strong clumping; as k → ∞ the distribution tends to the Poisson. It arises as a Poisson whose mean varies from unit to unit according to a gamma distribution (Chapter 4), which is the natural model for pests whose density differs from plant to plant.
The hypergeometric distribution describes sampling without replacement: from a lot of N items with M defectives, draw n; the number of defectives X has P(X = x) = C(M, x)C(N − M, n − x)/C(N, n). Its mean is nM/N, the same as the binomial with p = M/N, but its variance is n(M/N)(1 − M/N)(N − n)/(N − 1), smaller by the finite population correction (N − n)/(N − 1). As N → ∞ with M/N fixed, it tends to the binomial. Acceptance sampling of seed or fertiliser lots and the exact test for 2 × 2 tables are built on it.
The continuous uniform distribution on (a, b) has density 1/(b − a), mean (a + b)/2 and variance (b − a)²/12. It models rounding error and is the starting point of simulation through the probability integral transform.
The normal distribution N(μ, σ²) has density (σ√(2π))⁻¹ exp(−(x − μ)²/(2σ²)). It is symmetric and bell-shaped; mean, median and mode coincide at μ; the points of inflexion are at μ ± σ. All odd central moments are zero and μ₂ᵣ = 1·3·5⋯(2r − 1)σ²ʳ, so μ₄ = 3σ⁴ and β₂ = 3. The mean deviation about the mean is σ√(2/π) ≈ 0.7979σ and the quartile deviation ≈ 0.6745σ, giving the old rule QD : MD : SD ≈ 10 : 12 : 15. The MGF is exp(μt + σ²t²/2). About 68.27%, 95.45% and 99.73% of the distribution lies within 1, 2 and 3 standard deviations of the mean; ±1.96σ covers 95% and ±2.576σ covers 99%.
Linear combinations of independent normals are normal: if Xᵢ ~ N(μᵢ, σᵢ²), Σaᵢ Xᵢ ~ N(Σaᵢμᵢ, Σaᵢ²σᵢ²). Standardisation Z = (X − μ)/σ reduces every normal probability to the standard normal table. The normal is the limit of standardised sums (central limit theorem) and the error model behind least squares, analysis of variance and most of experimental design. Plot yields are often close to normal; counts and percentages usually need transformation first.
The exponential distribution with rate λ has density λe^(−λx), x > 0, mean 1/λ, variance 1/λ² (so its coefficient of variation is 1) and MGF λ/(λ − t) for t < λ. It is the only continuous distribution with the memoryless property P(X > s + t | X > s) = P(X > t), so its hazard rate is constant. It models times between random events (a Poisson process) and lifetimes of items that fail by chance rather than wear.
The gamma distribution with shape α and rate β has density βᵅxᵅ⁻¹e^(−βx)/Γ(α), mean α/β, variance α/β² and MGF (1 − t/β)^(−α). Some texts use a scale parameter θ = 1/β instead; check the parameterisation before using a formula. Gamma variables with a common rate add, with shapes adding. The sum of α independent exponentials with rate β is gamma with integer shape α (the Erlang distribution), and chi-square with n degrees of freedom is gamma with shape n/2 and rate 1/2. Gamma densities fit positively skewed quantities such as rainfall amounts on wet days. Γ(α) = (α − 1)! for integer α, Γ(1/2) = √π and Γ(α + 1) = αΓ(α).
The Weibull distribution, F(x) = 1 − exp[−(x/η)ᵏ] for x > 0 with shape k and scale η, has hazard rate proportional to xᵏ⁻¹: decreasing for k < 1 (early failures), constant for k = 1 (the exponential) and increasing for k > 1 (wear-out). Its mean is ηΓ(1 + 1/k). It is the standard model for life of machinery components and for wind-speed distributions, and it appears in extreme-value theory as the limiting law of minima.
The beta distribution of the first kind on (0, 1) has density x^(a−1)(1 − x)^(b−1)/B(a, b), with B(a, b) = Γ(a)Γ(b)/Γ(a + b). Its mean is a/(a + b) and variance ab/[(a + b)²(a + b + 1)]. It is uniform when a = b = 1, symmetric when a = b, U-shaped when both are below 1. It models proportions (the fraction of a field infested, germination rate across lots) and is the conjugate prior for a binomial p. The beta of the second kind, density x^(a−1)/[B(a, b)(1 + x)^(a+b)] on (0, ∞), is the distribution of X/Y for independent gammas with common scale, and the F distribution is a rescaled form of it.
The Cauchy distribution with location θ and scale σ has density σ/[π(σ² + (x − θ)²)]. It has no mean and no higher moments; θ is its median and mode, and θ ± σ are its quartiles. The mean of n independent Cauchy variables has the same Cauchy distribution as one observation, so averaging does not help, while the sample median is consistent for θ. It arises as the ratio of two independent standard normals and is Student's t with one degree of freedom.
X is lognormal if log X ~ N(μ, σ²). Its median is e^μ, its mean exp(μ + σ²/2), its mode exp(μ − σ²) and its variance (e^(σ²) − 1)exp(2μ + σ²); so mode < median < mean, a positive skew that grows with σ. All moments exist but the MGF does not. Products of many independent positive factors are approximately lognormal (the central limit theorem applied to logarithms), which is why farm sizes, incomes, pesticide residues and particle sizes in soils are often lognormal. Analysing such data on the log scale and back-transforming gives the median, not the mean.
The standard families form a connected web, and much of examination work is recognising which relation applies.
Sums: Bernoulli → binomial; geometric → negative binomial; exponential → gamma; Poisson + Poisson → Poisson; normal + normal → normal; chi-square + chi-square → chi-square (degrees of freedom add); gamma + gamma with a common rate → gamma. Special cases: the exponential is gamma with shape 1 and Weibull with shape 1; chi-square with 2 degrees of freedom is exponential with mean 2; the geometric is the negative binomial with r = 1; the uniform is beta(1, 1); the Cauchy is t₁.
Transformations: if X ~ N(0, 1), X² ~ χ²₁; if U ~ uniform(0, 1), −log U is exponential with mean 1; the ratio of independent standard normals is Cauchy; X/(X + Y) for independent gammas with a common scale is beta; Z/√(V/n) with V ~ χ²ₙ independent of Z is t; (U/m)/(V/n) for independent chi-squares is F(m, n); t²ₙ ~ F(1, n); and if F ~ F(m, n) then (mF/n)/(1 + mF/n) ~ beta(m/2, n/2). If X ~ F(m, n), then 1/X ~ F(n, m), which is why lower F percentage points are reciprocals of upper ones with the degrees of freedom swapped.
Conditioning and processes: in a Poisson process the waiting times between events are independent exponentials and the time to the k-th event is gamma (Erlang). The link between their CDFs, P(Poisson(λ) ≥ k) = P(Gamma(k, 1) ≤ λ), lets Poisson tail sums be read from gamma or chi-square tables. Similarly, binomial tail sums equal incomplete beta integrals.
Binomial to Poisson: if n → ∞ and p → 0 with np = λ fixed, C(n, x)pˣqⁿ⁻ˣ → e^(−λ)λˣ/x!. The approximation is good for large n and small p (a common working rule is n ≥ 20 with p ≤ 0.05, though texts differ). It is the law of rare events: the number of off-type plants in a large seed sample with a small off-type rate.
Binomial to normal (De Moivre-Laplace): (X − np)/√(npq) → N(0, 1) as n → ∞ for fixed p, used with a continuity correction; working rules ask for np and nq both reasonably large (5 or 10, depending on the text). Poisson to normal: (X − λ)/√λ → N(0, 1) as λ → ∞. Hypergeometric to binomial as N → ∞ with M/N fixed. Negative binomial to Poisson as k → ∞ with mean fixed. Gamma to normal as the shape grows.
Sampling distributions also have normal limits: tₙ → N(0, 1) as n → ∞ (the t table's last row is the normal); χ²ₙ is approximately N(n, 2n), and Fisher's approximation treats √(2χ²) as N(√(2n − 1), 1); F(m, n) → χ²_m/m as n → ∞. Fisher's z = (1/2) log F is closer to normal than F itself.
These limits matter in two ways. Practically, they let one use one table for many distributions. Theoretically, they explain why the normal and Poisson are so common: the normal is the limit of sums of many small effects, the Poisson the limit of many rare independent events. All of them are proved most easily by showing that the MGFs or characteristic functions converge (Lévy's continuity theorem).
A finite mixture has density f(x) = Σ πᵢ fᵢ(x), with weights πᵢ ≥ 0 summing to 1. It describes a population made of sub-populations: yields from two varieties grown unknowingly as a mixture, or fish lengths from two age classes. If component i has mean μᵢ and variance σᵢ², the mixture has mean Σπᵢμᵢ and variance Σπᵢσᵢ² + Σπᵢ(μᵢ − μ̄)², the within-plus-between decomposition of the law of total variance. A two-component normal mixture with well-separated means is bimodal.
A compound (or continuous mixture) distribution lets a parameter vary according to a mixing distribution. The most important case in agriculture: if a count X given λ is Poisson(λ), and λ varies from plant to plant as a gamma with shape k and rate β, the unconditional distribution of X is negative binomial with r = k and p = β/(1 + β), mean k/β and variance k/β + k/β². The extra-Poisson variation is exactly the variance of the mixing distribution. In the same way a binomial whose p varies as a beta(a, b) gives the beta-binomial, the standard model for over-dispersed proportions such as the number of diseased seedlings per tray when trays differ in infection pressure. Its variance is np(1 − p)[1 + (n − 1)ρ], with p = a/(a + b) and ρ = 1/(a + b + 1).
The word compound is also used for random sums, S = X₁ + ... + X_N with N random. A Poisson number of clusters, each containing a Poisson number of individuals, gives the Neyman Type A distribution, used for larvae hatched from egg masses. A Poisson number of claims of random size gives the compound Poisson model of crop-insurance losses. Their means and variances come from E(S) = E(N)E(X) and Var(S) = E(N)Var(X) + Var(N)[E(X)]².
A zero-inflated Poisson (ZIP) is a mixture of a point mass at zero, with weight π, and a Poisson(λ): P(0) = π + (1 − π)e^(−λ), P(x) = (1 − π)e^(−λ)λˣ/x! for x ≥ 1. Its mean is (1 − π)λ and variance (1 − π)λ(1 + πλ), so it is over-dispersed. It suits counts where some units cannot have the event at all (fields where the pest has not arrived) and others follow a Poisson.
A truncated distribution is the conditional distribution of X given that it falls in a set A: f_T(x) = f(x)/P(X ∈ A) for x in A. Truncation differs from censoring: in truncation, values outside A are not observed at all and their number is unknown; in censoring the number is known but the exact values are not.
The zero-truncated Poisson arises when only units with at least one event are recorded (only infested plants are examined, say): P(x) = e^(−λ)λˣ/[x!(1 − e^(−λ))] for x = 1, 2, ..., with mean λ/(1 − e^(−λ)). The zero-truncated binomial is analogous. For the normal truncated below at a, with α = (a − μ)/σ, the mean is μ + σφ(α)/[1 − Φ(α)]; the ratio φ(α)/[1 − Φ(α)] is the inverse Mills ratio, central to selection models in econometrics and to truncation selection in plant and animal breeding, where the mean of the selected parents exceeds the population mean by iσ with selection intensity i = φ(α)/[1 − Φ(α)]. For the exponential, truncation below at a simply shifts the distribution: by the memoryless property, X − a given X > a is the same exponential.
These distributions are named explicitly in AICE-JRF Unit 2. The examination tasks are to write the pmf or pdf, derive the mean and variance, and recognise the negative binomial and beta-binomial as mixtures.
Karl Pearson sought a single family that could represent most unimodal distributions met in practice. He defined it through the differential equation (1/f)(df/dx) = (x − a)/(b₀ + b₁x + b₂x²), whose constants are functions of the first four moments. The solutions are classified by the roots of the quadratic in the denominator, which is equivalent to the value of the criterion κ = β₁(β₂ + 3)² / [4(4β₂ − 3β₁)(2β₂ − 3β₁ − 6)].
The main types: κ < 0 gives Type I, the beta distribution of the first kind (bounded at both ends, skewed in general). β₁ = 0 with β₂ < 3 gives Type II, a symmetric beta. β₁ = 0 and β₂ = 3 is the normal curve. β₁ = 0 with β₂ > 3 gives Type VII, the Student-t form. 0 < κ < 1 gives Type IV, unbounded and skewed. κ = 1 gives Type V, the inverse-gamma form. κ > 1 gives Type VI, the beta of the second kind, which includes the F distribution. When 2β₂ − 3β₁ − 6 = 0 (κ infinite) the solution is Type III, the gamma distribution, of which chi-square and exponential are special cases. Pearson also listed further transitional types not needed here.
The practical use, in the days before computers, was to compute β₁ and β₂ from data, read off the type and fit it by moments. The system still provides a quick diagnostic: the (β₁, β₂) plane shows where each standard distribution lies, and the normal is the single point (0, 3). Moment estimates of β₁ and β₂ are unstable in small samples, so modern practice prefers likelihood-based fitting and probability plots.
Fitting a distribution means estimating its parameters from data and computing expected frequencies for comparison with observed ones.
Binomial: with N samples of size n, estimate p by x̄/n; the expected frequency of x is N·C(n, x)p̂ˣq̂ⁿ⁻ˣ, conveniently built up from N q̂ⁿ with the recurrence. Poisson: estimate λ by x̄ (the maximum likelihood estimate as well as the moment estimate); expected frequencies are N e^(−x̄)x̄ˣ/x!, built from Ne^(−x̄) with the recurrence × x̄/(x + 1). Negative binomial: the moment estimate of the aggregation parameter is k̂ = x̄²/(s² − x̄), sensible only when s² > x̄; maximum likelihood is preferred for small k.
Goodness of fit is judged by Pearson's chi-square, Σ(O − E)²/E, referred to chi-square with (number of classes − 1 − number of parameters estimated) degrees of freedom; classes with small expected frequencies (below 5 is the usual rule) are pooled first and the degrees of freedom counted after pooling. A Poisson fitted with λ estimated from the data to 7 classes after pooling has 7 − 1 − 1 = 5 degrees of freedom. The index-of-dispersion test is a quick check for Poisson: (n − 1)s²/x̄ is approximately χ² with n − 1 degrees of freedom under randomness.
Example of interpretation: counts of aphids on 200 leaves with mean 1.5 and variance 4.2 have an index of dispersion of 2.8, far above 1. A Poisson fit will show too few zeros and too few large counts in the expected column; a negative binomial with k̂ = 1.5²/(4.2 − 1.5) ≈ 0.83 will fit much better and tells the entomologist that the aphids are strongly clumped, which affects sampling plans and economic thresholds.
To fit a normal distribution to grouped data, estimate μ and σ by the sample mean and standard deviation, compute the standardised class boundaries z = (boundary − x̄)/s, find the area of each class from the normal table and multiply by N. Chi-square then has (classes − 1 − 2) degrees of freedom. Gamma and Weibull fits to rainfall or lifetime data use moments or maximum likelihood; the method of moments for the gamma gives shape x̄²/s² and rate x̄/s². Probability plots (quantile-quantile plots) are the modern visual check: data quantiles plotted against theoretical quantiles fall on a straight line when the model fits. Fitted distributions feed directly into practical decisions: return periods of drought or flood rainfall, design capacity of irrigation canals, sampling plans for pests and expected losses for insurance. A fit is never proof that the model is right: several distributions can pass a goodness-of-fit test on a few hundred observations and still give very different tail probabilities, which are often what the decision depends on.
Let X₁, ..., Xₙ be a random sample from a continuous distribution with CDF F and density f, and let X₍₁₎ < X₍₂₎ < ... < X₍ₙ₎ be the ordered values. X₍₁₎ is the minimum, X₍ₙ₎ the maximum, and the sample median and range are functions of these order statistics.
The maximum: P(X₍ₙ₎ ≤ x) = F(x)ⁿ, density nF(x)ⁿ⁻¹f(x). The minimum: P(X₍₁₎ > x) = [1 − F(x)]ⁿ, density n[1 − F(x)]ⁿ⁻¹f(x). In general, X₍ᵣ₎ ≤ x exactly when at least r observations are ≤ x, so P(X₍ᵣ₎ ≤ x) = Σ from j = r to n of C(n, j)F(x)ʲ[1 − F(x)]ⁿ⁻ʲ, a binomial tail. Differentiating, or arguing that r − 1 values fall below x, one at x and n − r above, gives the density f₍ᵣ₎(x) = n!/[(r − 1)!(n − r)!] F(x)ʳ⁻¹[1 − F(x)]ⁿ⁻ʳ f(x).
For the uniform (0, 1) distribution, F(x) = x and X₍ᵣ₎ ~ beta(r, n − r + 1), with mean r/(n + 1) and variance r(n − r + 1)/[(n + 1)²(n + 2)]. The expected ordered values divide (0, 1) into n + 1 equal parts, the basis of plotting positions r/(n + 1) in probability plots. Because F(X) is uniform, F(X₍ᵣ₎) has this beta distribution for any continuous F, which is how distribution-free tolerance intervals and confidence intervals for quantiles are built.
For the exponential with rate λ, the minimum of n is exponential with rate nλ, and the spacings X₍ᵢ₎ − X₍ᵢ₋₁₎ are independent exponentials with rates (n − i + 1)λ (Rényi's representation). Hence E(X₍ₙ₎) = (1/λ)(1 + 1/2 + ... + 1/n): the expected life of the last of n pumps grows only like log n.
The joint density of all n order statistics is n! Πf(xᵢ) on x₁ < x₂ < ... < xₙ. Integrating out the others gives the joint density of X₍ᵣ₎ and X₍ₛ₎ (r < s) at x < y: n!/[(r − 1)!(s − r − 1)!(n − s)!] F(x)ʳ⁻¹[F(y) − F(x)]ˢ⁻ʳ⁻¹[1 − F(y)]ⁿ⁻ˢ f(x)f(y). The case r = 1, s = n gives the joint density of minimum and maximum, n(n − 1)[F(y) − F(x)]ⁿ⁻²f(x)f(y).
The range R = X₍ₙ₎ − X₍₁₎ follows by the Jacobian method. For the uniform (0, 1), R ~ beta(n − 1, 2), with density n(n − 1)rⁿ⁻²(1 − r) and mean (n − 1)/(n + 1). For normal samples the mean range is a known multiple d₂ of σ, tabulated by sample size; this is why control charts for quality estimate σ by R̄/d₂ and why the range was used as a quick measure of variability in field data.
The sample median of an odd sample of size n = 2m + 1 is X₍ₘ₊₁₎. For large n it is approximately normal with mean the population median ξ and variance 1/[4nf(ξ)²]. For a normal population f(ξ) = 1/(σ√(2π)), giving variance πσ²/(2n), about 1.57 times the variance of the mean: the median is less efficient for normal data, but far more robust to outliers and the right choice for Cauchy-like data.
Order statistics are everywhere in agricultural work: the highest daily rainfall of a season, the earliest flowering date in a plot, the weakest link in a chain of irrigation joints, the trimmed means used for robust yield estimation and the extreme-value distributions (Gumbel, Fréchet, Weibull) that describe limits of normalised maxima and are used for flood and drought design. AICE-JRF Unit 2 names order statistics explicitly, and GATE ST asks for densities and expectations of single order statistics and ranges.
(X, Y) has a bivariate normal distribution with means μ₁, μ₂, standard deviations σ₁, σ₂ and correlation ρ (|ρ| < 1) if its density is f(x, y) = [2πσ₁σ₂√(1 − ρ²)]⁻¹ exp{−Q/[2(1 − ρ²)]}, where Q = u² − 2ρuv + v², u = (x − μ₁)/σ₁ and v = (y − μ₂)/σ₂. Its contours of equal density are ellipses centred at (μ₁, μ₂); with ρ > 0 they tilt upward. The joint MGF is exp(μ₁t₁ + μ₂t₂ + (σ₁²t₁² + 2ρσ₁σ₂t₁t₂ + σ₂²t₂²)/2).
Key properties: each marginal is normal, X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²); every linear combination aX + bY is normal with mean aμ₁ + bμ₂ and variance a²σ₁² + 2abρσ₁σ₂ + b²σ₂²; and X and Y are independent if and only if ρ = 0, the one family in which zero correlation implies independence. The converse of the marginal property fails: two normal marginals do not guarantee joint normality (if Z is standard normal and S an independent random sign, Z and SZ are each normal but Z + SZ has an atom at zero). Q in the exponent, divided by (1 − ρ²), has a chi-square distribution with 2 degrees of freedom, so the ellipse Q/(1 − ρ²) = 5.99 contains 95% of the distribution.
For the standardised bivariate normal, P(X > 0, Y > 0) = 1/4 + arcsin(ρ)/(2π), a result due to Sheppard: 1/4 when ρ = 0, 1/2 when ρ → 1.
The conditional distribution of Y given X = x is normal with mean μ₂ + ρ(σ₂/σ₁)(x − μ₁) and variance σ₂²(1 − ρ²). Three facts follow. The regression of Y on X is exactly linear, with slope β = ρσ₂/σ₁. The conditional variance does not depend on x (homoscedasticity). And the proportion of variance in Y explained by X is ρ², the coefficient of determination. By symmetry, the regression of X on Y has slope ρσ₁/σ₂; the two regression lines differ unless |ρ| = 1, and the product of the two slopes is ρ².
A worked case: plant height X (cm) and grain yield Y (g per plant) are bivariate normal with μ₁ = 100, σ₁ = 10, μ₂ = 50, σ₂ = 8 and ρ = 0.6. For plants 110 cm tall, the conditional mean yield is 50 + 0.6 × (8/10) × 10 = 54.8 g and the conditional standard deviation is 8√(1 − 0.36) = 6.4 g. Selecting for height thus shifts yield by ρσ₂ per σ₁ of height, the same reasoning as correlated response in breeding.
Regression toward the mean is visible here: plants one standard deviation above average in height are, on average, only ρ standard deviations above average in yield. Galton found the same pattern in parents' and offspring heights, which is where the word regression comes from.
The bivariate normal extends to the multivariate normal Nₚ(μ, Σ), whose marginals, conditionals and linear transforms are all normal; the conditional mean is linear in the conditioning variables, and the partitioned-covariance formulas generalise those above. It is the model behind multiple regression, discriminant analysis, selection indices in breeding and multi-trait analysis of variance.
If Z₁, ..., Zₙ are independent standard normals, V = ΣZᵢ² has the chi-square distribution with n degrees of freedom: a gamma with shape n/2 and rate 1/2, density proportional to v^(n/2 − 1)e^(−v/2) on v > 0. Its mean is n, variance 2n, MGF (1 − 2t)^(−n/2) for t < 1/2, and mode n − 2 for n ≥ 2. It is positively skewed, less so as n grows; independent chi-squares add, degrees of freedom adding.
The central sampling result: for a random sample from N(μ, σ²), the sample mean X̄ and the sample variance s² = Σ(Xᵢ − X̄)²/(n − 1) are independent, X̄ ~ N(μ, σ²/n), and (n − 1)s²/σ² ~ χ²ₙ₋₁. One degree of freedom is lost because the deviations from X̄ sum to zero. The independence of X̄ and s² holds only for normal samples; it characterises the normal distribution.
Uses: confidence intervals and tests for a normal variance (is the variability of a new variety's yield within the limit claimed?), Pearson's goodness-of-fit and contingency-table statistics (asymptotically chi-square), likelihood-ratio tests (−2 log Λ is asymptotically chi-square), and Bartlett's test of homogeneity of variances before pooling error mean squares across experiments.
If Z ~ N(0, 1) and V ~ χ²ₙ are independent, T = Z/√(V/n) has Student's t distribution with n degrees of freedom. The density is proportional to (1 + t²/n)^(−(n+1)/2): symmetric about zero, bell-shaped, with heavier tails than the normal. Its mean is 0 for n > 1 and its variance n/(n − 2) for n > 2; in general moments of order r exist only for r < n. With n = 1 it is the Cauchy distribution; as n → ∞ it tends to N(0, 1). For n > 4, β₂ = 3 + 6/(n − 4).
Applied to a normal sample, (X̄ − μ)/(s/√n) ~ tₙ₋₁, because it is Z = (X̄ − μ)/(σ/√n) divided by √(s²/σ²) and the two are independent. This is the statistic of the one-sample and paired t-tests and of the confidence interval x̄ ± t(s/√n). The two-sample statistic with pooled variance has n₁ + n₂ − 2 degrees of freedom under equal variances. Gosset derived the distribution in 1908 while working on small-sample problems in the Guinness brewery's barley and hop trials.
If U ~ χ²_m and V ~ χ²ₙ are independent, F = (U/m)/(V/n) has the F distribution with (m, n) degrees of freedom. Its mean is n/(n − 2) for n > 2, which does not depend on m; its variance is 2n²(m + n − 2)/[m(n − 2)²(n − 4)] for n > 4; its mode, for m > 2, is [(m − 2)/m][n/(n + 2)], always below 1. Reciprocal property: 1/F ~ F(n, m), so F₁₋α(m, n) = 1/F_α(n, m), where F_α denotes the upper α point. Relations: t²ₙ ~ F(1, n); mF(m, n) → χ²_m as n → ∞; and mF/(n + mF) ~ beta(m/2, n/2).
The F distribution is the distribution of a variance ratio. For independent normal samples, (s₁²/σ₁²)/(s₂²/σ₂²) ~ F(n₁ − 1, n₂ − 1), used to compare variances. In analysis of variance the ratio of the treatment mean square to the error mean square is F with (treatment df, error df) under the null hypothesis of no treatment effects, because the two sums of squares are independent chi-squares (Chapter 11). Every F-test in a field-trial report is this result. Fisher worked with z = (1/2) log F, whose distribution is nearer normal; Snedecor tabulated F itself and named it after Fisher.
When the normal variables have non-zero means, the sum of their squares is no longer central chi-square. If Xᵢ ~ N(μᵢ, 1) independently, V = ΣXᵢ² has the non-central chi-square distribution with n degrees of freedom and non-centrality parameter λ = Σμᵢ². Texts differ: some define the non-centrality as λ/2 or as √λ, so read the definition before using a formula. With λ = Σμᵢ², the mean is n + λ and the variance 2(n + 2λ); the MGF is (1 − 2t)^(−n/2) exp[λt/(1 − 2t)].
The non-central chi-square is a Poisson mixture of central chi-squares: V has the same distribution as χ²ₙ₊₂J, where J is Poisson with mean λ/2. Independent non-central chi-squares add, with degrees of freedom and non-centralities both adding. The distribution depends on the μᵢ only through λ. When λ = 0 it reduces to the central chi-square. In the same normal-sample setting, if X₁, ..., Xₙ are N(μ, σ²), then ΣXᵢ²/σ² is non-central χ²ₙ with λ = nμ²/σ², while Σ(Xᵢ − X̄)²/σ² remains central χ²ₙ₋₁: centring removes the non-centrality.
If Z ~ N(0, 1) and V ~ χ²ₙ are independent and δ is a constant, T = (Z + δ)/√(V/n) has the non-central t distribution with n degrees of freedom and non-centrality δ. It is not symmetric when δ ≠ 0. If U is non-central chi-square with m degrees of freedom and non-centrality λ, independent of V ~ χ²ₙ, then F = (U/m)/(V/n) is non-central F(m, n, λ); its mean is n(m + λ)/[m(n − 2)] for n > 2, larger than the central mean. If the denominator is also non-central, the result is the doubly non-central F.
The role of these distributions is power. Under the null hypothesis a t or F statistic has its central distribution; under an alternative it has the non-central one, with non-centrality determined by the true effect, the error variance and the sample size. For the one-sample t-test δ = √n(μ − μ₀)/σ. For a one-way analysis of variance with t treatments and r replicates, λ = rΣτᵢ²/σ², with τᵢ the treatment effects. Power is the probability that the non-central variable exceeds the central critical value.
Planning the number of replicates for a field trial means choosing r so that the non-central F reaches the required power, often 0.8, for the smallest treatment difference worth detecting. Because λ grows linearly with r, doubling replication doubles the non-centrality; it does not double the power, which rises along an S-shaped curve. Tables and charts of the non-central F (Pearson-Hartley charts, Tang's tables) were the traditional tools; statistical software now computes non-central probabilities directly.
The practical lessons for a trial statistician are: an under-powered trial that finds 'no significant difference' has not shown that varieties are equal; reducing σ² through better blocking raises λ exactly as more replicates do; and the non-centrality parameter, not the p-value, is the right quantity to report when planning the next season's trial. AICE-JRF and GATE ST examine the definitions, means and the Poisson-mixture representation; ISS papers ask for the derivation of the MGF of the non-central chi-square.
A family of densities or mass functions is a one-parameter exponential family if it can be written f(x; θ) = h(x) exp[η(θ)T(x) − A(θ)], where h and T depend only on x, η and A only on θ, and the support {x : f(x; θ) > 0} does not depend on θ. η is the natural (canonical) parameter and T(x) the natural sufficient statistic. The k-parameter version has Σ ηⱼ(θ)Tⱼ(x) in the exponent.
Members, with their natural parameters: binomial with known n (η = log[p/(1 − p)], the logit, T = x); Poisson (η = log λ, T = x); geometric and negative binomial with known r (η = log q); normal with known variance (η = μ/σ², T = x); normal with both parameters unknown (two-parameter, T = (x, x²)); exponential and gamma with known shape (η = −β); gamma with both parameters (T = (x, log x)); beta (T = (log x, log(1 − x))); and the multinomial with fixed n.
Non-members: any family whose support depends on the parameter, such as uniform(0, θ) or the shifted exponential with unknown location; the Cauchy with unknown location; Student's t with unknown location; the Weibull with unknown shape; and most mixtures. The hypergeometric with unknown M is also not an exponential family.
The exponential family is where classical inference works best. For a sample of size n, ΣT(Xᵢ) is a sufficient statistic of fixed dimension whatever n is; it is complete when the natural parameter space contains an open interval (open set in k dimensions). Completeness plus sufficiency gives unique minimum-variance unbiased estimators through the Lehmann-Scheffé theorem. In the natural parameterisation the moments of T come from A: E[T(X)] = A′(η) and Var[T(X)] = A″(η), and A(η) is convex. For the Poisson, A(η) = e^η, so E(X) = Var(X) = e^η = λ.
If η(θ) is monotone, the family has a monotone likelihood ratio in T, so uniformly most powerful one-sided tests exist and reject for large (or small) ΣT(Xᵢ): the Karlin-Rubin theorem. The likelihood equation for the natural parameter sets the sample mean of T equal to its expectation, so maximum likelihood and the method of moments coincide for T. Each member has a conjugate prior: beta for binomial, gamma for Poisson, normal for a normal mean with known variance, which keeps Bayesian updating in closed form. Regularity conditions for the Cramér-Rao bound hold automatically.
Generalised linear models are built on the exponential family with a dispersion parameter: binomial with logit link for germination or mortality data, Poisson with log link for counts, gamma with log link for positive skewed yields. The canonical link is the one that sets the linear predictor equal to the natural parameter. Recognising whether a model belongs to the family is therefore not an academic exercise; it decides which estimation theory and which software routines apply.
Examination questions ask whether a given density belongs, what its natural parameter and sufficient statistic are, and why a parameter-dependent support excludes a family. A quick test: take logarithms; if log f can be written as a sum of a term in x alone, a term in θ alone and products of functions of θ with functions of x, and the support is fixed, the family belongs.
Analysis of variance splits a total sum of squares into components, and each component is a quadratic form X′AX in the observations, with A a symmetric matrix. The theory below explains why each component, divided by σ², is chi-square and why the components are independent, so that their ratios are F.
Let X ~ Nₙ(0, I). Then X′AX ~ χ²_r if and only if A is idempotent (A² = A); its degrees of freedom are r = rank(A) = trace(A). If X ~ Nₙ(μ, I), X′AX with A idempotent is non-central chi-square with r degrees of freedom and non-centrality μ′Aμ. For X ~ Nₙ(0, Σ), X′AX is chi-square if and only if AΣ is idempotent (AΣA = A), with degrees of freedom rank(A) when Σ is non-singular. The expectation needs no normality: E(X′AX) = trace(AΣ) + μ′Aμ for any X with mean μ and covariance Σ. For normal X the variance is 2 trace(AΣAΣ) + 4μ′AΣAμ.
Example: the sample sum of squares Σ(Xᵢ − X̄)² = X′(I − J/n)X, where J is the n × n matrix of ones. The centring matrix I − J/n is symmetric and idempotent with trace n − 1, so for a normal sample with variance σ², Σ(Xᵢ − X̄)²/σ² ~ χ²ₙ₋₁. The mean X̄ = (1/n)1′X is a linear form.
For X ~ Nₙ(0, σ²I): two quadratic forms X′AX and X′BX are independent if and only if AB = 0. A linear form LX and a quadratic form X′AX are independent if and only if LA = 0. For general covariance Σ the conditions become AΣB = 0 and LΣA = 0. Applied to the example: (1/n)1′(I − J/n) = (1/n)(1′ − 1′) = 0, so X̄ and Σ(Xᵢ − X̄)² are independent, the result behind Student's t.
A second example shows how the condition is checked in a design. In a randomised block layout, the treatment and block sums of squares are quadratic forms whose matrices are orthogonal projections onto the treatment and block contrasts. Because every treatment appears once in every block, these contrast spaces are orthogonal, the product of the two matrices is zero, and the two sums of squares are independent. When a plot is missing, orthogonality is lost, the product is no longer zero, and that is precisely why missing-plot formulas or a general linear-model analysis are needed. Non-orthogonality is the matrix form of 'the effects are confounded with each other'.
Cochran's theorem (1934): let X ~ Nₙ(0, I) and suppose X′X = Q₁ + Q₂ + ... + Q_k, where Qᵢ = X′AᵢX with ranks rᵢ. Then the Qᵢ are independent chi-squares with rᵢ degrees of freedom if and only if r₁ + r₂ + ... + r_k = n. Equivalent conditions are that each Aᵢ is idempotent, or that AᵢAⱼ = 0 for all i ≠ j. With non-zero means the Qᵢ are independent non-central chi-squares. Some texts call the matrix version the Fisher-Cochran theorem.
This is the theorem that makes analysis of variance work. In a one-way layout with t treatments and r replicates (n = rt plots), the corrected total sum of squares, with n − 1 degrees of freedom, splits into the treatment sum of squares, rank t − 1, and the error sum of squares, rank t(r − 1). The ranks add: (t − 1) + t(r − 1) = n − 1. Under normal, independent, equal-variance errors and no treatment effects, SS(treatment)/σ² and SS(error)/σ² are independent chi-squares, and F = [SS(treatment)/(t − 1)]/[SS(error)/(t(r − 1))] ~ F(t − 1, t(r − 1)). With treatment effects present, the treatment sum of squares becomes non-central with λ = rΣτᵢ²/σ², which gives the power of the test (Chapter 9). The same argument covers randomised blocks, Latin squares, factorial components and regression sums of squares: degrees of freedom add, the matrices are orthogonal projections, and the F-ratios follow.
The assumptions are what the theorem needs: normality, independence and a common variance. When counts or percentages violate them, the transformations of the probability manual (square root, arcsine, log) are applied first, or a generalised linear model is used instead. The examiner's favourite tasks are: decide whether a given matrix is idempotent and find its rank by the trace, show independence of X̄ and s², find E(X′AX), and verify the rank condition for an ANOVA partition.
A student uses this subject to recognise distributions on sight and to derive what is asked: the mean and variance of a negative binomial, the distribution of a sum from its MGF, the density of the third smallest of five observations, the conditional mean in a bivariate normal, or why the ANOVA F-ratio follows an F distribution. In GATE ST, IIT JAM MS and AICE-JRF most distribution questions reduce to knowing the relations in Chapters 3 and 7 and the order-statistic formula in Chapter 6. A thesis that analyses counts, proportions or lifetimes needs the same knowledge to choose a model before choosing a test.
A statistician five years into work at a research station or plant-protection service fits distributions to data every week: testing whether pest counts are Poisson or over-dispersed and estimating the negative binomial k for sampling plans, choosing a beta-binomial or a binomial GLM for germination data, fitting gamma or Weibull distributions to rainfall and machinery lifetimes, using order statistics for extreme rainfall and robust trimmed means, and computing the replication needed for a trial from non-central F. Explaining to scientists why their counts violate ANOVA assumptions, and what to do about it, is part of the job.
A senior statistician with twenty-five years of experience sets the distributional assumptions behind institutional methods: the count models used in national pest surveillance, the extreme-value models behind drought and flood return periods in crop insurance, and the standard power calculations required before multi-location trials are approved. Such a statistician recognises where standard models fail in practice, through zero inflation, mixtures of sub-populations and truncation by the way data are collected, and insists on models that match the sampling process. Reviewing other scientists' analyses for these errors is a large part of the role.
A PhD with long experience works on distribution theory itself or uses it to build methods: deriving exact and asymptotic distributions of new test statistics through quadratic-form theory and Cochran-type decompositions, developing compound and mixture models for over-dispersed and zero-inflated agricultural counts, extending order-statistic and extreme-value theory to spatially dependent field data, and using exponential-family structure for generalised linear mixed models in multi-environment trials. Such a researcher also supervises students through these derivations and judges when a neat closed form is less useful than simulation.
Derivations are sketched rather than given in full. Pearson Types VIII to XII, the Johnson and Burr systems, generalised extreme-value theory, multivariate distributions beyond the bivariate normal (Wishart, Hotelling's T²), circular distributions and copulas are not covered; the multivariate normal in depth belongs to Multivariate Analysis. Working rules for when Poisson or normal approximations are adequate differ between textbooks, so the manual says so rather than fixing one. Non-centrality parameters are defined in more than one way across texts; the manual states its convention (λ = Σμᵢ²). Statistical tables (normal, t, chi-square, F, d₂ for ranges) are not reproduced; questions state the values they need.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed with a script; the values needed are stated in the question.
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