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Previous year question hub

Convergence of Random Variables - Statistics Previous Year Questions

Practice Convergence of Random Variables - Statistics previous year questions organised from real papers, with year-wise coverage and clear topic navigation.

8Papers
8Years
26Questions
1Topics

Convergence of Random Variables question pattern

Every graph below is calculated only from this selection.

Questions by year

Year-wise coverage for Convergence of Random Variables. Each bar uses a separate theme-derived color.

Difficulty distribution

How the classified questions are distributed by difficulty.

Medium 19 73.1%
Hard 5 19.2%
Easy 2 7.7%

Question type distribution

MCQ, numerical, multiple-select and other formats found in these papers.

Numerical Answer Type (NAT) 9 34.6%
MCQ 9 34.6%
MSQ 8 30.8%

Subject weightage

Top subjects by unique question coverage.

Statistics
26 Qs

Most asked topics

Top topics across the included previous year papers.

Convergence of Random Variables
26 Qs

Subtopic coverage

Top subtopics inside this exact selection.

Modes of Convergence
14 Qs
Laws of Large Numbers and Central Limit Theorem
12 Qs

Paper coverage

Question coverage for the most populated papers. Every active PYP paper remains listed below.

Statistics (ST) 2026
2 Qs
Statistics (ST) 2025
4 Qs
Statistics (ST) 2024
4 Qs
Statistics (ST) 2023
5 Qs
Statistics (ST) 2022
3 Qs
Statistics (ST) 2021
4 Qs
Statistics (ST) 2020
1 Qs
Statistics (ST) 2019
3 Qs

Browse by subtopics

Open a focused page built from the same verified paper data.

Modes of Convergence

Explore previous-paper coverage, trends and focused practice for Modes of Convergence.

14 questions6 papers6 years

Laws of Large Numbers and Central Limit Theorem

Explore previous-paper coverage, trends and focused practice for Laws of Large Numbers and Central Limit Theorem.

12 questions8 papers8 years

Included previous year papers

Newest papers appear first. Sort by year, question coverage or name.

PaperYear / sessionQuestions in this viewOpen
Statistics (ST) 202620262View paper
Statistics (ST) 202520254View paper
Statistics (ST) 202420244View paper
Statistics (ST) 202320235View paper
Statistics (ST) 202220223View paper
Statistics (ST) 202120214View paper
Statistics (ST) 202020201View paper
Statistics (ST) 201920193View paper

Sample previous year questions

A varied preview from the papers represented in this selection, with every available option.

1
2019 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem
Statistics (ST) 2019
Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed normal random variables with mean 4 and variance 1. Then \( \lim_{n \to \infty} P \left( \frac{1}{n} \sum_{i=1}^n X_i > 4.0006 \right) \) is equal to ...
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2
2020 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem
Statistics (ST) 2020
Let \(\{X_n\}_{n \ge 1}\) be a sequence of independent and identically distributed random variables with \(E(X_1) = 0, E(X_1^2) = 1\) and \(E(X_1^4) = 3\). Further, let \[ Y_n = \frac{X_1^2 + X_2^2 + \dots + X_n^2}{n}. \] If \[ \lim_{n \to \infty} P\left( Y_n + \frac{\sqrt{n}(Y_n - 1)}{\sqrt{3}} \le 2 \right) = \Phi(c), \] where \(\Phi(\cdot)\) denotes the cumulative distribution function of the standard normal distribution, then \(c^2\) = ____________________ (correct up to one decimal place).
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3
2021 · Statistics · Convergence of Random Variables · Modes of Convergence
Statistics (ST) 2021
For $\alpha > 0$, let $\{X_n^{(\alpha)}\}_{n=1}^{\infty}$ be a sequence of independent random variables such that \(P(X_n^{(\alpha)} = 1) = \frac{1}{n^{2\alpha}} = 1 - P(X_n^{(\alpha)} = 0).\) Let $S = \{ \alpha > 0 : X_n^{(\alpha)} \text{ converges to } 0 \text{ almost surely as } n \to \infty \}$. Then the infimum of $S$ equals ________ (round off to 2 decimal places).
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4
2022 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem
Statistics (ST) 2022
Let \(X_i\), \(i = 1, 2, ..., n\), be i.i.d. random variables with the probability density function \(f_X(x) = \begin{cases} \frac{1}{\sqrt{2} \Gamma(\frac{1}{2})} x^{-5/6} e^{-x/8}, & 0 < x < \infty, \\ 0, & \text{elsewhere}, \end{cases}\) where \(\Gamma(\cdot)\) denotes the gamma function. Also, let \(\bar{X}_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n)\). If \(\sqrt{n} (\bar{X}_n (3 - \bar{X}_n) - \frac{20}{n})\) converges to \(N(0, \sigma^2)\) in distribution, then \(\sigma^2\) (rounded off to two decimal places) is equal to ______
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5
2023 · Statistics · Convergence of Random Variables · Modes of Convergence
Statistics (ST) 2023
Let \(\{X_n\}_{n \geq 1}\) and \(\{Y_n\}_{n \geq 1}\) be two sequences of random variables and \(X\) and \(Y\) be two random variables, all of them defined on the same probability space. Which one of the following statements is true?
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6
2024 · Statistics · Convergence of Random Variables · Modes of Convergence
Statistics (ST) 2024
Let \( \{F_n\}_{n \geq 1} \) be a sequence of cumulative distribution functions given by
\[ F_n(x) = \begin{cases} 0 & \text{if } x < -n \\ \frac{x + n}{2n} & \text{if } -n \leq x < n \\ 1 & \text{if } x \geq n. \end{cases} \]
Which one of the following statements is true?
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