CUET-PG Probability Theory: practice questions

CUET-PG - Statistics · India · National Testing Agency

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

10worked here, free

85written to this exam in this subject

11syllabus units

What this subject covers

Probability from its axioms to the limit theorems: random variables, expectation, generating functions, inequalities and modes of convergence, ending with the laws of large numbers and the central limit theorem that justify most of applied statistics.

In the exams: AICE-JRF Ag Stat Unit 1 (heavy); GATE ST 'Probability' and limit theorems (heavy); IIT JAM MS (heavy); UPSC ISS Statistics I (heavy); ISI admission (heavy); CSIR-NET Mathematical Sciences Unit 4 (present)

Sample spaces, sigma-fields and the axioms of probability
Conditional probability, independence and Bayes' theorem
Random variables, distribution functions, mass and density functions
Expectation, moments and moment-generating, probability-generating and characteristic functions
Joint, marginal and conditional distributions; conditional expectation
Functions of random variables and transformations (Jacobian method)
Inequalities: Markov, Chebyshev, Jensen and Cauchy-Schwarz
Modes of convergence and their relations; Slutsky's lemma; Borel-Cantelli lemma
Weak and strong laws of large numbers
Central limit theorems (Lindeberg-Levy, Liapounov) and the delta method
Elements of measure theory: measurable functions and Lebesgue integral

What CUET-PG asks from it

ICAR AIEEA-PG Group 07, CUET-PG Statistics, SSC JSO and state statistical services

Classical and conditional probability, addition and multiplication rules, Bayes' theorem, expectation and variance, Chebyshev's inequality and the normal approximation.

Where the 85 questions fall

How our questions for CUET-PG spread across the units of Probability Theory. Difficulty: 35 easy, 42 medium, 8 hard.

Sample spaces, sigma-fields and the axioms of probability	10
Expectation, moments and moment-generating, probability-generating and characteristic functions	10
Inequalities: Markov, Chebyshev, Jensen and Cauchy-Schwarz	9
Conditional probability, independence and Bayes' theorem	8
Random variables, distribution functions, mass and density functions	8
Joint, marginal and conditional distributions; conditional expectation	8
Functions of random variables and transformations (Jacobian method)	8
Modes of convergence and their relations; Slutsky's lemma; Borel-Cantelli lemma	7
Central limit theorems (Lindeberg-Levy, Liapounov) and the delta method	7
Weak and strong laws of large numbers	6
Elements of measure theory: measurable functions and Lebesgue integral	4

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 · Conditional probability, independence and Bayes' theorem · medium

A test for a seed-borne virus has sensitivity 0.95 and specificity 0.9. The virus is present in 2% of lots. A lot tests positive. What is the probability that it is infected?

0.162
0.950
0.100
0.019

Answer and working

A. 0.162

Bayes: P(infected | +) = (0.02 × 0.95)/(0.02 × 0.95 + 0.98 × 0.10) = 0.019/0.117 = 0.162. 0.950 is the sensitivity P(+ | infected), the commonest confusion; 0.019 is the joint probability P(infected and +), not the conditional.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

2 · Random variables, distribution functions, mass and density functions · easy

X is uniform on (0, 0.5), so its density equals 2 on that interval. Which statement is correct?

This is invalid: a uniform density on any interval must equal 1 everywhere on it
This is valid: a density must integrate to 1, and its values are allowed to exceed 1
This is invalid: every density value is a probability and so lies between 0 and 1
This is valid only because the interval (0, 0.5) is open rather than closed

Answer and working

B. This is valid: a density must integrate to 1, and its values are allowed to exceed 1

A pdf must be non-negative and integrate to 1; 2 × 0.5 = 1, so it is a valid density. Density values are not probabilities: P(X = x) = 0 for a continuous variable, and probabilities come from areas.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

3 · Expectation, moments and moment-generating, probability-generating and characteristic functions · medium

Which statement about a characteristic function φ(t) = E(e^(itX)) is NOT true for every random variable X?

φ is uniformly continuous on the real line
φ(−t) is the complex conjugate of φ(t)
|φ(t)| ≤ 1 for every real value of t
φ(t) is a real number for every value of t

Answer and working

D. φ(t) is a real number for every value of t

φ(0) = 1, |φ| ≤ 1, uniform continuity and φ(−t) = conj(φ(t)) hold for all distributions. φ is real-valued only when the distribution is symmetric about 0; an exponential variable, for example, has φ(t) = 1/(1 − it), which is complex.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

4 · Joint, marginal and conditional distributions; conditional expectation · medium

(X, Y) has joint density f(x, y) = x + y on 0 < x < 1, 0 < y < 1. What is the marginal density of X?

2x for 0 < x < 1
x + 1/2 for 0 < x < 1
x + y for 0 < x < 1
x for 0 < x < 1

Answer and working

B. x + 1/2 for 0 < x < 1

f_X(x) = ∫₀¹ (x + y) dy = x + 1/2. It integrates to 1/2 + 1/2 = 1 as it must. A marginal cannot contain y; dropping the y-integral gives x, which integrates to only 1/2.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

5 · Functions of random variables and transformations (Jacobian method) · medium

Z is standard normal and Y = Z². Which is the density of Y for y > 0?

(1/2)(2πy)^(−1/2) e^(−y/2)
(1/2) e^(−y/2)
(2π)^(−1/2) e^(−y²/2)
(2πy)^(−1/2) e^(−y/2)

Answer and working

D. (2πy)^(−1/2) e^(−y/2)

F_Y(y) = P(−√y ≤ Z ≤ √y) = 2Φ(√y) − 1; differentiating gives 2φ(√y) × 1/(2√y) = (2πy)^(−1/2)e^(−y/2), the chi-square(1) density. The version with the extra 1/2 counts only one branch. (1/2)e^(−y/2) is chi-square with 2 df.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

6 · Inequalities: Markov, Chebyshev, Jensen and Cauchy-Schwarz · medium

Plant height has mean 50 cm and standard deviation 4 cm; the distribution is unknown. What does Chebyshev's inequality give as an upper bound for P(|X − 50| ≥ 10)?

0.40
0.84
0.0124
0.16

Answer and working

D. 0.16

P(|X − μ| ≥ c) ≤ σ²/c² = 16/100 = 0.16. 0.40 uses σ/c without squaring; 0.84 is the lower bound for the complement; 0.0124 is the exact value only if X were normal.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

7 · Modes of convergence and their relations; Slutsky's lemma; Borel-Cantelli lemma · medium

Xₙ converges in distribution to a constant c. What can be concluded?

Xₙ converges to c in mean square as well
Xₙ converges to c in probability as well
Xₙ converges to c almost surely as well
Nothing more, since the limit is only in law

Answer and working

B. Xₙ converges to c in probability as well

For a degenerate limit, P(|Xₙ − c| > ε) ≤ 1 − Fₙ(c + ε) + Fₙ(c − ε) → 0, because c ± ε are continuity points. Almost-sure and mean-square convergence need more (moments may not even exist).

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

8 · Weak and strong laws of large numbers · medium

For iid X₁, X₂, ..., Kolmogorov's strong law states that X̄ₙ converges almost surely to a finite constant if and only if:

Var(X₁) > 0, and then the constant is the median
the Xᵢ are bounded, and then the constant is 0
E(X₁²) < ∞, and then the constant is E(X₁)
E|X₁| < ∞, and then the constant is E(X₁)

Answer and working

D. E|X₁| < ∞, and then the constant is E(X₁)

Kolmogorov's SLLN: X̄ₙ → μ a.s. iff E|X₁| < ∞, with μ = E(X₁). A finite second moment is sufficient but not necessary; boundedness is far stronger than needed.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

9 · Weak and strong laws of large numbers · easy

What is the essential difference between the weak and the strong law of large numbers?

The weak law holds for small samples; the strong law holds only for samples over thirty
The weak law needs a finite variance; the strong law needs only identically distributed variables
The weak law gives convergence in probability; the strong law gives almost-sure convergence
The weak law concerns sums; the strong law concerns the normal limit of sums

Answer and working

C. The weak law gives convergence in probability; the strong law gives almost-sure convergence

The laws differ in the mode of convergence of X̄ₙ to μ. Khintchine's WLLN needs no variance, the SLLN needs E|X| < ∞, and neither is about a particular sample size or about normality, which is the CLT.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

10 · Central limit theorems (Lindeberg-Levy, Liapounov) and the delta method · medium

√n(Tₙ − θ) converges in distribution to N(0, σ²), and g′(θ) = 0 while g″(θ) ≠ 0. Which statement is correct?

√n(g(Tₙ) − g(θ)) → 0 in probability, and n(g(Tₙ) − g(θ)) has a scaled chi-square(1) limit
√n(g(Tₙ) − g(θ)) → N(0, σ²), the same limit as for Tₙ itself
No limit exists, and the delta method cannot be adapted to this case
√n(g(Tₙ) − g(θ)) → N(0, g″(θ)²σ²), with g″ replacing g′

Answer and working

A. √n(g(Tₙ) − g(θ)) → 0 in probability, and n(g(Tₙ) − g(θ)) has a scaled chi-square(1) limit

First-order delta method gives variance g′(θ)²σ² = 0, a degenerate limit. The second-order expansion g(Tₙ) − g(θ) ≈ g″(θ)(Tₙ − θ)²/2 gives n(g(Tₙ) − g(θ)) → (g″(θ)σ²/2)χ²₁.

Further reading: Rohatgi and Saleh, An Introduction to Probability and Statistics; Casella and Berger, Statistical Inference

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