IIT JAM Real Analysis for Statistics: practice questions

IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)

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

10worked here, free

39written 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)

Sets, countability and the real number system
Sequences: convergence, monotone and Cauchy sequences
Series: tests of convergence, absolute and conditional convergence
Power series and radius of convergence
Limits, continuity and uniform continuity
Differentiability, Rolle's and mean value theorems, Taylor's theorem
L'Hospital's rule and indeterminate forms
Riemann integration and improper integrals
Gamma and beta functions
Functions of several variables: partial derivatives, Jacobians, maxima, minima and Lagrange multipliers
Double and triple integrals and change of variables

Where the 39 questions fall

How our questions for IIT JAM spread across the units of Real Analysis for Statistics. Difficulty: 6 easy, 27 medium, 6 hard.

Sequences: convergence, monotone and Cauchy sequences	4
Power series and radius of convergence	4
L'Hospital's rule and indeterminate forms	4
Gamma and beta functions	4
Functions of several variables: partial derivatives, Jacobians, maxima, minima and Lagrange multipliers	4
Double and triple integrals and change of variables	4
Sets, countability and the real number system	3
Series: tests of convergence, absolute and conditional convergence	3
Limits, continuity and uniform continuity	3
Differentiability, Rolle's and mean value theorems, Taylor's theorem	3
Riemann integration and improper integrals	3

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 · Sequences: convergence, monotone and Cauchy sequences · hard

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

The normal approximation to the t distribution
The variance of the sample mean
The central limit theorem for proportions
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)

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

What is Σₙ₌₁^∞ 1/(n(n + 2))?

Answer and working

A. 3/4

1/(n(n + 2)) = ½(1/n − 1/(n + 2)); the sum telescopes to ½(1 + ½) = 3/4. 1 is the sum of 1/(n(n + 1)).

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

For which series is the ratio test inconclusive?

Σ1/2ⁿ
Σn!/nⁿ
Σ3ⁿ/n!
Σ1/n²

Answer and working

D. Σ1/n²

For any p-series the ratio tends to 1. The others give ratios 1/2, 1/e and 0.

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

What is Σₙ₌₁^∞ n/2ⁿ?

Answer and working

B. 2

Σnrⁿ = r/(1 − r)² = (1/2)/(1/4) = 2.

5 · Riemann integration and improper integrals · medium

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

p < 1
p > 1
p ≤ 1
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.

6 · Gamma and beta functions · hard

What is ∫₀^∞ x² e^(−3x) dx?

2/9
1/27
2/27
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!.

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

For f(x, y) = x² + y² − 2x − 4y, the critical point and its nature are:

(1, 2), a maximum with value 5
(1, 2), a minimum with value −5
(2, 4), a saddle point
(−1, −2), a minimum with value −5

Answer and working

B. (1, 2), a minimum with value −5

f_x = 2x − 2, f_y = 2y − 4 give (1, 2); D = 4 > 0, f_xx > 0; f(1, 2) = 1 + 4 − 2 − 8 = −5.

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

For f(x, y) = x³ − 3x + y², the point (−1, 0) is:

A local minimum
A local maximum
Not a critical point
A saddle point

Answer and working

D. A saddle point

f_xx = 6x = −6, f_yy = 2, f_xy = 0, so D = −12 < 0: saddle. (1, 0) is the minimum.

9 · Double and triple integrals and change of variables · medium

For u = x + y and v = x − y, the absolute value of ∂(x, y)/∂(u, v) is:

2
1
−2
1/2

Answer and working

D. 1/2

∂(u, v)/∂(x, y) = −2, so its reciprocal has absolute value 1/2.

10 · Double and triple integrals and change of variables · medium

If (U, V) is a one-to-one transformation of (X, Y), the joint density of (U, V) is:

f_X,Y(x(u, v), y(u, v)) × |∂(x, y)/∂(u, v)|
f_X,Y(u, v) with no adjustment
f_X,Y(x, y) × |∂(u, v)/∂(x, y)|
The product of the marginal densities of X and Y

Answer and working

A. f_X,Y(x(u, v), y(u, v)) × |∂(x, y)/∂(u, v)|

The density is evaluated at the inverse transformation and multiplied by the absolute Jacobian of the inverse.

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