CUET-PG - Statistics · India · National Testing Agency
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 AIEEA-PG Group 07, CUET-PG Statistics, SSC JSO and state statistical services
Binomial, Poisson and normal: mean, variance, mode, fitting to data, normal-area calculations and the chi-square, t and F tests that follow.
How our questions for CUET-PG spread across the units of Probability Distributions & Sampling Distributions. Difficulty: 29 easy, 33 medium, 4 hard.
| Continuous distributions: uniform, normal, exponential, gamma, beta, Cauchy, Weibull, lognormal | 10 | |
| Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial | 9 | |
| Sampling distributions: chi-square, t and F and their properties | 9 | |
| Relations and limiting forms among distributions | 7 | |
| Bivariate normal: marginal and conditional distributions | 7 | |
| Pearsonian system and fitting of distributions to agricultural data | 6 | |
| Exponential family of distributions | 6 | |
| Order statistics: distributions of single, joint and range | 5 | |
| Compound, mixture and truncated distributions | 3 | |
| Distributions of quadratic forms in normal variables (Cochran's theorem) | 3 | |
| Non-central chi-square, t and F | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
A binomial variable has mean 6 and variance 2.4. What is n?
B. 10
q = variance/mean = 2.4/6 = 0.4, so p = 0.6 and n = 6/0.6 = 10. 15 comes from taking p = 0.4 (confusing p with q); 6 confuses n with the mean.
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
X is uniform on (2, 8). What is Var(X)?
A. 3
Var = (b − a)²/12 = 36/12 = 3. 5 is the mean; 36 forgets to divide by 12.
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
Calls to a farmers' helpline in a day are Poisson with mean 100. Using the normal approximation with continuity correction, what is P(at least 120 calls)?
B. 0.0256
Mean = variance = 100, SD 10. P(X ≥ 120) ≈ P(Z > (119.5 − 100)/10) = P(Z > 1.95) = 0.0256. Without correction z = 2 gives 0.0228; dividing by the variance gives z = 0.2.
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
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 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
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
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