CSIR-UGC NET Probability Distributions & Sampling Distributions: practice questions

CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR

Original practice questions written to the published syllabus · not past papers

6worked here, free

30written to this exam in this subject

9syllabus units

What this subject covers

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)

Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial
Continuous distributions: uniform, normal, exponential, gamma, beta, Cauchy, Weibull, lognormal
Relations and limiting forms among distributions
Compound, mixture and truncated distributions
Pearsonian system and fitting of distributions to agricultural data
Order statistics: distributions of single, joint and range
Bivariate normal: marginal and conditional distributions
Sampling distributions: chi-square, t and F and their properties
Non-central chi-square, t and F
Exponential family of distributions
Distributions of quadratic forms in normal variables (Cochran's theorem)

Where the 30 questions fall

How our questions for CSIR-UGC NET spread across the units of Probability Distributions & Sampling Distributions. Difficulty: 3 easy, 10 medium, 17 hard.

Non-central chi-square, t and F	8
Distributions of quadratic forms in normal variables (Cochran's theorem)	7
Exponential family of distributions	6
Bivariate normal: marginal and conditional distributions	3
Relations and limiting forms among distributions	2
Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial	1
Pearsonian system and fitting of distributions to agricultural data	1
Order statistics: distributions of single, joint and range	1
Sampling distributions: chi-square, t and F and their properties	1

6 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial · hard

X and Y are independent Poisson variables with means 2 and 3. What is P(X = 2 | X + Y = 4)?

0.3750
0.4000
0.3456
0.2707

Answer and working

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

2 · Non-central chi-square, t and F · medium

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?

Answer and working

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.

3 · Exponential family of distributions · hard

For a random sample from a beta(a, b) distribution with both parameters unknown, what is the natural sufficient statistic?

(Σ log Xᵢ, Σ log(1 − Xᵢ))
(ΣXᵢ, ΣXᵢ²)
(Π Xᵢ alone)
(ΣXᵢ, Σ(1 − Xᵢ))

Answer and working

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.

4 · Distributions of quadratic forms in normal variables (Cochran's theorem) · medium

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?

trace(LA) = n
AL = I
L′L = A
LA = 0

Answer and working

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.

5 · Distributions of quadratic forms in normal variables (Cochran's theorem) · hard

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?

Their sum is χ²₂, but neither is chi-square
Each is χ²₂, and they are independent
Each is χ²₁, and they are independent
Each is χ²₁, but they are dependent

Answer and working

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.

6 · Distributions of quadratic forms in normal variables (Cochran's theorem) · hard

X ~ Nₙ(μ, I) and A is symmetric idempotent of rank r. What is the distribution of X′AX?

Normal with mean μ′Aμ and variance 2r
Central chi-square with r df, whatever μ
Non-central chi-square with n df and non-centrality μ′μ
Non-central chi-square with r df and non-centrality μ′Aμ

Answer and working

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.

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