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GATE Real Analysis for Statistics: practice questions

GATE - Statistics (ST) · India · IITs and IISc on rotation (2026: IIT Guwahati)

Original practice questions written to the published syllabus · not past papers
10worked here, free
41written to this exam in this subject
11syllabus units

What this subject covers

The rigorous calculus behind limits theorems and estimation: sequences, series, continuity, differentiability and Riemann integration, at the level GATE Statistics and JAM Mathematical Statistics examine and that measure-theoretic probability later builds on.

In the exams: GATE ST 'Calculus' section (heavy); IIT JAM MS mathematics component (heavy); AIEEA-PG Group 07 Unit II-III mathematics (present); ISI admission tests (heavy); UPSC ISS Statistics I (light)

Where the 41 questions fall

How our questions for GATE spread across the units of Real Analysis for Statistics. Difficulty: 2 easy, 26 medium, 13 hard.

Functions of several variables: partial derivatives, Jacobians, maxima, minima and Lagrange multipliers5
Series: tests of convergence, absolute and conditional convergence4
Power series and radius of convergence4
Differentiability, Rolle's and mean value theorems, Taylor's theorem4
L'Hospital's rule and indeterminate forms4
Riemann integration and improper integrals4
Gamma and beta functions4
Double and triple integrals and change of variables4
Sequences: convergence, monotone and Cauchy sequences3
Limits, continuity and uniform continuity3
Sets, countability and the real number system2

10 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 · Sets, countability and the real number system · medium

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.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

2 · Sequences: convergence, monotone and Cauchy sequences · hard

The limit (1 − λ/n)ⁿ → e^(−λ) is used in statistics to derive:

  1. The normal approximation to the t distribution
  2. The variance of the sample mean
  3. The central limit theorem for proportions
  4. The Poisson approximation to the binomial
Answer and working

D. The Poisson approximation to the binomial

P(X = 0) = (1 − λ/n)ⁿ → e^(−λ) when p = λ/n; the same argument gives every Poisson probability.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

3 · Series: tests of convergence, absolute and conditional convergence · medium

What is Σₙ₌₁^∞ n/2ⁿ?

  1. 1
  2. 2
  3. 4
  4. 1/2
Answer and working

B. 2

Σnrⁿ = r/(1 − r)² = (1/2)/(1/4) = 2.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

4 · Power series and radius of convergence · hard

The moment generating function of a distribution, where it exists near 0, has the expansion M(t) = Σ E(Xⁿ)tⁿ/n!. The coefficient of t² is therefore:

  1. E(X²)
  2. Var(X)
  3. E(X)²/2
  4. E(X²)/2
Answer and working

D. E(X²)/2

The n = 2 term is E(X²)t²/2!; Var(X) needs E(X)² as well.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

5 · Limits, continuity and uniform continuity · medium

A cumulative distribution function can have discontinuities only of which kind?

  1. Essential, and uncountably many
  2. Removable only
  3. None, because CDFs are always continuous
  4. Jumps, and at most countably many
Answer and working

D. Jumps, and at most countably many

CDFs are monotone, and monotone functions have only jump discontinuities, countably many.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

6 · Differentiability, Rolle's and mean value theorems, Taylor's theorem · medium

What is the coefficient of x³ in the Maclaurin series of e^(2x)?

  1. 8
  2. 2/3
  3. 1/6
  4. 4/3
Answer and working

D. 4/3

e^(2x) = Σ(2x)ⁿ/n!; the x³ coefficient is 8/3! = 8/6 = 4/3.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

7 · Riemann integration and improper integrals · medium

∫₀¹ x^(−p) dx converges if and only if:

  1. p < 1
  2. p > 1
  3. p ≤ 1
  4. p = 1 only
Answer and working

A. p < 1

The antiderivative x^(1−p)/(1 − p) stays finite at 0 when p < 1; at p = 1 it is ln x, which diverges.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

8 · Riemann integration and improper integrals · medium

What is d/dx of ∫₀^(x²) eᵗ dt?

  1. e^(x²)
  2. x² e^(x²)
  3. 2x e^(x²)
  4. 2e^(x²)
Answer and working

C. 2x e^(x²)

Leibniz rule: f(b(x))b′(x) = e^(x²) × 2x.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

9 · Gamma and beta functions · hard

What is ∫₀^∞ x² e^(−3x) dx?

  1. 2/9
  2. 1/27
  3. 2/27
  4. 6/27
Answer and working

C. 2/27

Γ(3)/3³ = 2/27; 2/9 uses 3² and 6/27 uses 3! instead of 2!.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

10 · Functions of several variables: partial derivatives, Jacobians, maxima, minima and Lagrange multipliers · medium

The minimum of x² + y² subject to x + y = 4 is:

  1. 16
  2. 4
  3. 2
  4. 8
Answer and working

D. 8

Lagrange gives x = y = 2, so x² + y² = 8.

Further reading: Rudin, Principles of Mathematical Analysis (1976); Bartle and Sherbert, Introduction to Real Analysis (2011); Apostol, Mathematical Analysis (1974); Casella and Berger, Statistical Inference (2002)

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Real Analysis for StatisticsThe full study manual, free GATEFormat, sections and how it has changed

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