GATE - Statistics (ST)

India · IITs and IISc on rotation (2026: IIT Guwahati) · pg

902practice questions written to this syllabus
16subjects

Format

Computer-based, 3 hours, 100 marks (15 General Aptitude + 85 subject). MCQ, MSQ and NAT; negative marking on MCQ only; on-screen virtual calculator.

Sections

Over the years

Paper introduced in GATE 2021. Used for MSc/MTech/PhD admission at IITs and some PSU and research-institute recruitment. The subject list above is read from the 2026 syllabus; the overall GATE pattern (marks split, marking scheme) is common to all papers.

Official source: https://gate2026.iitg.ac.in/doc/GATE2026_Syllabus/ST_2026_Syllabus.pdf

Subjects this exam draws on

Real Analysis for Statistics41 questionsLinear Algebra & Matrix Theory for Statistics37 questionsDifferential Equations, Finite Differences & Numerical Analysis3 questionsProbability Theory77 questionsProbability Distributions & Sampling Distributions61 questionsStochastic Processes86 questionsStatistical Inference I: Estimation84 questionsStatistical Inference II: Testing of Hypotheses & Sequential Analysis61 questionsNonparametric & Rank-Based Methods65 questionsLinear Models & Regression Analysis28 questionsMultivariate Analysis76 questionsCategorical Data Analysis & Generalized Linear Models74 questionsBayesian Inference71 questionsBioassay, Survival & Epidemiological Statistics62 questionsOptimization Techniques for Statisticians40 questionsStatistical Quality Control & Reliability36 questions

Five practice questions, worked

Original questions written to this exam's published syllabus and pattern. Not past papers.

1. The set of irrational numbers is:

  1. Countable, like the rationals
  2. Finite and bounded
  3. Uncountable
  4. Empty in any bounded interval
Answer and working

C. Uncountable

R = Q ∪ irrationals; if the irrationals were countable, R would be a union of two countable sets and hence countable, which it is not.

Real Analysis for Statistics · Sets, countability and the real number system

2. What is the dimension of the space of all 3 × 3 real symmetric matrices?

  1. 9
  2. 6
  3. 3
  4. 5
Answer and working

B. 6

n(n + 1)/2 = 6 free entries (diagonal plus upper triangle). 3 is the skew-symmetric dimension.

Linear Algebra & Matrix Theory for Statistics · Vector spaces, subspaces, linear independence, basis and dimension

3. The solution of dy/dt + 2y = 4 with y(0) = 0 is:

  1. y = 4(1 − e^(−2t))
  2. y = 2e^(−2t)
  3. y = 4t
  4. y = 2(1 − e^(−2t))
Answer and working

D. y = 2(1 − e^(−2t))

Equilibrium y = 4/2 = 2; y = 2 + Ce^(−2t) with y(0) = 0 gives C = −2.

Differential Equations, Finite Differences & Numerical Analysis · Ordinary differential equations of first order: exact, linear and Bernoulli equations

4. A collection 𝓕 of subsets of Ω contains Ω. Which further pair of closure properties makes 𝓕 a σ-field?

  1. Closure under complements and under finite unions only
  2. Closure under complements and under countable unions
  3. Closure under arbitrary unions, countable or uncountable
  4. Closure under finite intersections, with no condition on complements
Answer and working

B. Closure under complements and under countable unions

A σ-field needs Ω, complements and countable unions; countable intersections then follow by De Morgan. Closure under complements and finite unions only defines a field (algebra), which is weaker. Arbitrary uncountable unions are not required, and without complements the collection need not be a σ-field.

Probability Theory · Sample spaces, sigma-fields and the axioms of probability

5. A geometric variable X counts the failures before the first success in independent trials with success probability 0.25. What is E(X)?

  1. 3
  2. 0.75
  3. 4
  4. 12
Answer and working

A. 3

For the failures-before-first-success version, E(X) = q/p = 0.75/0.25 = 3. The trials-until-success version has mean 1/p = 4; 12 is the variance q/p².

Probability Distributions & Sampling Distributions · Discrete distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric and multinomial

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