ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR
Original practice questions written to the published syllabus · not past papersEvery 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)
ICAR AICE-JRF/SRF Agricultural Statistics (Units 1 and 2)
Properties of standard distributions, relations among them, compound and truncated distributions, order statistics, sampling distributions and their properties; recognising the negative binomial as a Poisson-gamma mixture and the role of Cochran's theorem in ANOVA.
How our questions for ICAR AICE-JRF/SRF spread across the units of Probability Distributions & Sampling Distributions. Difficulty: 7 easy, 20 medium, 21 hard.
| Non-central chi-square, t and F | 9 | |
| Distributions of quadratic forms in normal variables (Cochran's theorem) | 8 | |
| Compound, mixture and truncated distributions | 7 | |
| Exponential family of distributions | 7 | |
| Bivariate normal: marginal and conditional distributions | 5 | |
| Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial | 3 | |
| Relations and limiting forms among distributions | 3 | |
| Pearsonian system and fitting of distributions to agricultural data | 3 | |
| Order statistics: distributions of single, joint and range | 2 | |
| Sampling distributions: chi-square, t and F and their properties | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
X and Y are independent Poisson variables with means 2 and 3. What is P(X = 2 | X + Y = 4)?
C. 0.3456
Given X + Y = 4, X ~ binomial(4, 2/5): P(X = 2) = C(4, 2)(0.4)²(0.6)² = 0.3456. 0.2707 is the unconditional P(X = 2); 0.3750 uses p = 1/2, ignoring the unequal means.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
Insects per plant are Poisson with mean λ, and λ varies from plant to plant according to a gamma distribution. What is the unconditional distribution of insects per plant?
D. Negative binomial
Integrating e^(−λ)λˣ/x! against a gamma(k, β) density gives C(x + k − 1, x)(β/(1 + β))ᵏ(1/(1 + β))ˣ: negative binomial with r = k. The variance exceeds the mean by the variance of the mixing distribution, so it cannot be Poisson.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
Plant height in a population is normal with mean 100 cm and standard deviation 10 cm. The taller half is selected (truncation at the mean). What is the mean height of the selected plants?
A. 107.98 cm
Truncated below at a = μ, α = 0: mean = μ + σφ(0)/[1 − Φ(0)] = 100 + 10 × 0.3989/0.5 = 107.98 cm (selection intensity i = 0.798). Forgetting to divide by the proportion selected gives 103.99 cm.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
For a random sample of size n from a continuous distribution with CDF F and density f, which is the density of the r-th order statistic X₍ᵣ₎?
D. n!/[(r − 1)!(n − r)!] F(x)^(r−1) [1 − F(x)]^(n−r) f(x)
r − 1 observations fall below x, one at x and n − r above: the multinomial count n!/[(r − 1)! 1! (n − r)!] times F^(r−1) f (1 − F)^(n−r). The C(n, r)Fʳ(1 − F)ⁿ⁻ʳ form is one term of the CDF sum, not a density.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
Plant height X (cm) and grain yield Y (g) are bivariate normal with μ_X = 100, σ_X = 10, μ_Y = 50, σ_Y = 8 and ρ = 0.6. What is the mean yield of plants 110 cm tall?
D. 54.8 g
E(Y | X = x) = μ_Y + ρ(σ_Y/σ_X)(x − μ_X) = 50 + 0.6 × 0.8 × 10 = 54.8 g. Using σ_X/σ_Y inverts the slope and gives 57.5 g; 58.0 g assumes perfect correlation.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
A one-way trial has 3 varieties with true effects −2, 0 and 2 kg, 4 replicates each and error variance 4 kg². With λ = rΣτᵢ²/σ², what is the non-centrality parameter of the treatment F-statistic?
C. 8
λ = 4 × (4 + 0 + 4)/4 = 8. Omitting r gives 2; dividing by σ = 2 instead of σ² gives 16; using |τ| instead of τ² gives 4.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
For a random sample from a beta(a, b) distribution with both parameters unknown, what is the natural sufficient statistic?
A. (Σ log Xᵢ, Σ log(1 − Xᵢ))
log f = (a − 1) log x + (b − 1) log(1 − x) − log B(a, b), so T = (log x, log(1 − x)). Note ΣXᵢ and Σ(1 − Xᵢ) carry the same information and are not sufficient here. ΠXᵢ alone covers only a.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
X ~ Nₙ(0, σ²I), A is symmetric and L is a matrix. Under what condition are the linear form LX and the quadratic form X′AX independent?
D. LA = 0
LX and X′AX are independent iff LA = 0. With L = (1/n)1′ and A = I − J/n, LA = 0, which proves the independence of X̄ and Σ(Xᵢ − X̄)² for normal samples.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
X₁ and X₂ are iid N(0, 1). What are the distributions of (X₁ + X₂)²/2 and (X₁ − X₂)²/2, and are they independent?
C. Each is χ²₁, and they are independent
(X₁ ± X₂)/√2 are N(0, 1) and uncorrelated, hence independent under joint normality; their squares are independent χ²₁. In matrix terms A = J/2 and B = I − J/2 are idempotent of rank 1 with AB = 0, and the ranks add to 2.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
X ~ Nₙ(μ, I) and A is symmetric idempotent of rank r. What is the distribution of X′AX?
D. Non-central chi-square with r df and non-centrality μ′Aμ
Write A = PP′ with P′P = I_r; then P′X ~ N_r(P′μ, I) and X′AX = ‖P′X‖² is non-central χ²_r with λ = μ′PP′μ = μ′Aμ. It is central only when Aμ = 0.
Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Johnson, Kotz and Balakrishnan, Continuous Univariate Distributions; Johnson, Kemp and Kotz, Univariate Discrete Distributions
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