IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)
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)
GATE Statistics (ST) and IIT JAM Mathematical Statistics
Numerical and multiple-select items: moments from MGFs, identifying distributions of sums, ratios and transformations, densities and expectations of order statistics, bivariate normal conditionals, chi-square, t and F relations, and exponential-family membership.
How our questions for IIT JAM spread across the units of Probability Distributions & Sampling Distributions. Difficulty: 17 easy, 26 medium, 17 hard.
| Order statistics: distributions of single, joint and range | 10 | |
| Bivariate normal: marginal and conditional distributions | 8 | |
| Sampling distributions: chi-square, t and F and their properties | 8 | |
| Continuous distributions: uniform, normal, exponential, gamma, beta, Cauchy, Weibull, lognormal | 6 | |
| Exponential family of distributions | 6 | |
| Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial | 5 | |
| Relations and limiting forms among distributions | 5 | |
| Compound, mixture and truncated distributions | 5 | |
| Non-central chi-square, t and F | 3 | |
| Distributions of quadratic forms in normal variables (Cochran's theorem) | 3 | |
| Pearsonian system and fitting of distributions to agricultural data | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Wet-day rainfall X is gamma with shape 3 and rate 0.5 per mm, so f(x) ∝ x² e^(−0.5x). What is Var(X)?
C. 12 mm²
With rate β, mean = α/β = 6 mm and variance = α/β² = 3/0.25 = 12 mm². 6 is the mean; 0.75 = αβ² wrongly treats 0.5 as a scale parameter (a parameterisation slip).
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 has a Cauchy distribution with location 5 and scale 2. What is its interquartile range?
A. 4
The Cauchy quartiles are θ ± σ = 3 and 7, so the IQR is 2σ = 4. The Cauchy has no mean or variance, but its quantiles exist. 1.349σ is the IQR of a normal.
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
V has a chi-square distribution with 2 degrees of freedom. What is P(V > 6)?
B. 0.0498
χ²₂ is exponential with mean 2: P(V > 6) = e^(−6/2) = e^(−3) = 0.0498. e^(−6) treats the mean as 1.
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, ..., X₄ are independent exponential variables with rate 2. What is the distribution of their sum?
A. Gamma with shape 4 and rate 2, mean 2
The MGF of the sum is [2/(2 − t)]⁴, the gamma (Erlang) MGF with shape 4 and rate 2: mean 4/2 = 2. Rate 8 is the distribution of the minimum of the four, not the sum.
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
A large sample of n = 100 plot yields is drawn from a normal population with σ = 10. What is the approximate standard error of the sample median?
D. 1.2533
Var(median) ≈ 1/[4nf(ξ)²] with f(ξ) = 1/(σ√(2π)), giving πσ²/(2n) = 1.5708; SE = 1.2533. 1.0000 is the standard error of the mean, σ/√n; 1.5708 is the variance, not the standard error.
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 random sample of 16 is drawn from N(μ, 4). What is the variance of the sample variance s² (divisor n − 1)?
B. 2.1333
(n − 1)s²/σ² ~ χ²₁₅ with variance 30, so Var(s²) = σ⁴ × 2(n − 1)/(n − 1)² = 2σ⁴/(n − 1) = 32/15 = 2.1333. 30 is the variance of the chi-square itself; dividing by n gives 2.0000.
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 has the F distribution with (5, 12) degrees of freedom. What is E(X)?
B. 1.2
E(F) = n/(n − 2) for denominator df n > 2: 12/10 = 1.2, regardless of the numerator df 5. 1.67 = 5/3 uses the numerator df.
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
Ten here, 60 in the app. The Laisha app has every practice question we have written to IIT JAM in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.
Free for ten days, then one subscription of US$10 a month. We would rather say so here than have you find it later.
An AI operating system for agriculture: free guides, manuals and tools written for every country.