Modes of Convergence - Convergence of Random Variables - Statistics Previous Year Questions

1

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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2

2021 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2021

Let \( \{X_n\}_{n \ge 1} \) be a sequence of independent and identically distributed random variables having common distribution function \( F(\cdot) \). Let \( a < b \) be two real numbers such that \( F(x) = 0 \) for all \( x \le a \), \( 0 < F(x) < 1 \) for all \( a < x < b \), and \( F(x) = 1 \) for all \( x \ge b \). Let \( S_n(x) \) be the empirical distribution function at \( x \) based on \( X_1, X_2, ..., X_n, n \ge 1 \). Then which one of the following statements is NOT true?

A

\(P\left[ \lim_{n \to \infty} \sup_{-\infty < x < \infty} |S_n(x) - F(x)| = 0 \right] = 1 \)

B

For fixed \(x \in (a, b) \) and \(t \in (-\infty, \infty) \), \(\lim_{n \to \infty} P\left[ \frac{\sqrt{n} \ [S_n(x) - F(x)]}{\sqrt{S_n(x)(1 - S_n(x))}} \le t \right] = P(Z \le t) \), where \(Z \) is the standard normal random variable

C

The covariance between \(S_n(x) \) and \(S_n(y) \) equals \(\frac{1}{n} F(x)(1 - F(y)) \) for all \(n \ge 2 \) and for fixed \(-\infty < x, y < \infty \)

D

If \(Y_n = \sup_{-\infty < x < \infty} (S_n(x) - F(x))^2 \), then \(\{4n Y_n\}_{n \ge 1} \) converges in distribution to a central chi-square random variable with \(2 \) degrees of freedom

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3

2021 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2021

Let \(\{X_n\}_{n \geq 1}\) be a sequence of independent and identically distributed random variables each having probability density function
\(f(x) = \begin{cases} e^{-x}, & x > 0, \\ 0, & \text{otherwise}. \end{cases}\).
Let \(X_{(n)} = \max\{X_1, X_2, ..., X_n\}\) for \(n \geq 1\). If \(Z\) is the random variable to which \(\{X_{(n)} - \log_e n\}_{n \geq 1}\) converges in distribution, as \(n \to \infty\), then the median of \(Z\) equals __________ (round off to 2 decimal places).

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4

2022 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2022

Suppose \( X_1, X_2, ..., X_n, ... \) are independent exponential random variables with the mean \( \frac{1}{2} \). Let the notation \( i.o. \) denote 'infinitely often'. Then which of the following is/are true?

A

\(P\left(\left\{X_n > \frac{\epsilon}{2} \log_e n\right\} \text{ i.o.}\right) = 1 \) for \(0 < \epsilon \leq 1 \)

B

\(P\left(\left\{X_n < \frac{\epsilon}{2} \log_e n\right\} \text{ i.o.}\right) = 1 \) for \(0 < \epsilon \leq 1 \)

C

\(P\left(\left\{X_n > \frac{\epsilon}{2} \log_e n\right\} \text{ i.o.}\right) = 1 \) for \(\epsilon > 1 \)

D

\(P\left(\left\{X_n < \frac{\epsilon}{2} \log_e n\right\} \text{ i.o.}\right) = 1 \) for \(\epsilon > 1 \)

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5

2022 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2022

Let \(\{X_n\}, n \geq 1\), be a sequence of random variables with the probability mass functions
\(p_{X_n}(x) = \begin{cases} \frac{n}{n+1}, & x = 0, \\ \frac{1}{n+1}, & x = n, \\ 0, & \text{elsewhere}. \end{cases}\)
Let \(X\) be a random variable with \(P(X = 0) = 1\). Then which of the following statements is/are true?

A

\(X_n\) converges to \(X\) in distribution

B

\(X_n\) converges to \(X\) in probability

C

\(E(X_n) \to E(X)\)

D

There exists a subsequence \(\{X_{n_k}\}\) of \(\{X_n\}\) such that \(X_{n_k}\) converges to \(X\) almost surely

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6

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?

A

If \(\{X_n\}_{n \geq 1}\) converges in distribution to a real constant \(c\), then \(\{X_n\}_{n \geq 1}\) converges in probability to \(c\)

B

If \(\{X_n\}_{n \geq 1}\) converges in probability to \(X\), then \(\{X_n\}_{n \geq 1}\) converges in \(3^{rd}\) mean to \(X\)

C

If \(\{X_n\}_{n \geq 1}\) converges in distribution to \(X\) and \(\{Y_n\}_{n \geq 1}\) converges in distribution to \(Y\), then \(\{X_n + Y_n\}_{n \geq 1}\) converges in distribution to \(X + Y\)

D

If \(\{E(X_n)\}_{n \geq 1}\) converges to \(E(X)\), then \(\{X_n\}_{n \geq 1}\) converges in \(1^{st}\) mean to \(X\)

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7

2023 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2023

For a given real number \( a \), let \( a^+ = max\{a, 0\} \) and \( a^- = max\{-a, 0\} \). If \( \{x_n\}_{n \geq 1} \) is a sequence of real numbers, then which of the following statements is/are true?

A

If \(\{x_n\}_{n \geq 1} \) converges, then both \(\{x_n^+\}_{n \geq 1} \) and \(\{x_n^-\}_{n \geq 1} \) converge

B

If \(\{x_n\}_{n \geq 1} \) converges to 0, then both \(\{x_n^+\}_{n \geq 1} \) and \(\{x_n^-\}_{n \geq 1} \) converge to 0

C

If both \(\{x_n^+\}_{n \geq 1} \) and \(\{x_n^-\}_{n \geq 1} \) converge, then \(\{x_n\}_{n \geq 1} \) converges

D

If \(\{x_n^2\}_{n \geq 1} \) converges, then both \(\{x_n^+\}_{n \geq 1} \) and \(\{x_n^-\}_{n \geq 1} \) converge

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8

2023 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2023

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables with mean \( 0 \) and variance \( 1 \), all of them defined on the same probability space. For \( n = 1, 2, 3, ... \), let \( Y_n = \frac{1}{n} (X_1 X_2 + X_3 X_4 + \cdots + X_{2n-1} X_{2n}) \). Then which of the following statements is/are true?

A

\(\{\sqrt{n} Y_n\}_{n \geq 1} \) converges in distribution to a standard normal random variable

B

\(\{Y_n\}_{n \geq 1} \) converges in \(2^{nd} \) mean to \(0 \)

C

\(\{Y_n + \frac{1}{n}\}_{n \geq 1} \) converges in probability to \(0 \)

D

\(\{X_n\}_{n \geq 1} \) converges almost surely to \(0 \)

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9

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?

A

\(F_n(x) \) converges for all \(x \in \mathbb{R} \) and the limiting function is a cumulative distribution function

B

\(F_n(x) \) converges for all \(x \in \mathbb{R} \), but the limiting function is not a cumulative distribution function

C

\(F_n(x) \) does not converge for any \(x \in \mathbb{R} \)

D

There exist \(x, y \in \mathbb{R} \) such that \(F_n(x) \) converges but \(F_n(y) \) does not converge

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10

2024 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2024

Let \(\{X_n\}_{n \geq 1}\) be a sequence of independent random variables such that \(X_n\) has Poisson distribution with mean \(\lambda_n\), where \(\lambda_n = \lambda + \frac{1}{2^n}\), \(n \geq 1\), and \(\lambda > 0\) is an unknown parameter. Which one of the following statements is true?

A

\(\frac{1}{n} \sum_{i=1}^n X_i\) is an unbiased estimator of \(\lambda\)

B

\(\frac{1}{n} \sum_{i=1}^n X_i\) is a consistent estimator of \(\lambda\)

C

\(\sum_{i=1}^n X_i\) is a consistent estimator of \(\lambda\)

D

\(\frac{1}{n^2} \sum_{i=1}^n X_i\) is an unbiased estimator of \(\lambda\)

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11

2024 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2024

Let \(\{X_n\}_{n \ge 1}\) be a sequence of independent random variables such that the probability density function of \(X_n\) is given by
\[f_n(x) = \begin{cases} \frac{1}{\lambda_n} e^{-\frac{x}{\lambda_n}} & \text{if } x \ge 0 \\ 0 & \text{otherwise,} \end{cases}\]
where \(\lambda_n = 10 - \sum_{i=1}^n \frac{5}{2^{i-1}}\). Which of the following statements is/are true?

A

\(\{X_n\}_{n \ge 1}\) converges in distribution to the zero random variable

B

\(\{X_n\}_{n \ge 1}\) converges in probability to the zero random variable

C

\(\{X_n\}_{n \ge 1}\) converges in distribution to a random variable having Poisson distribution with mean 10

D

\(\{X_n\}_{n \ge 1}\) converges in probability to a random variable having Poisson distribution with mean 10

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12

2025 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2025

Let \(\{X_n\}_{n \geq 1}\) be a sequence of i.i.d. random variables with common distribution function \(F\), and let \(F_n\) be the empirical distribution function based on \(\{X_1, X_2, ..., X_n\}\). Then, for each fixed \(x \in (-\infty, \infty)\), which one of the following options is correct?

A

\(\sqrt{n}(F_n(x) - F(x)) \xrightarrow{p} 0\) as \(n \to \infty\)

B

\(\frac{n(F_n(x) - F(x))}{\sqrt{F(x)(1 - F(x))}} \xrightarrow{d} Z\) as \(n \to \infty\), where \(Z \sim N(0,1)\)

C

\(F_n(x) \xrightarrow{a.s.} F(x)\) as \(n \to \infty\)

D

\(\lim_{n \to \infty} n \operatorname{Var}(F_n(x)) = 0\)

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13

2025 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2025

Let \(\{X_n\}_{n \geq 1}\) be a sequence of independent random variables and \(X_n \xrightarrow{a.s.} 0\) as \(n \to \infty\). Then which of the following options is/are necessarily correct?

A

\(E(X_n^2) \to 0\) as \(n \to \infty\)

B

\(X_n^2 \xrightarrow{p} 0\) as \(n \to \infty\)

C

For any \(\epsilon > 0\), \(\sum_{n=1}^{\infty} \Pr(|X_n| \geq \epsilon) < \infty\)

D

\(X_n^2 + X_n + 5 \xrightarrow{a.s.} 5\) as \(n \to \infty\)

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14

2026 · Statistics · Convergence of Random Variables · Modes of Convergence

Statistics (ST) 2026

Let \( \{X_n\}_{n \geq 1} \) be a sequence of random variables having the following probability mass function \[ P(X_n = x) = \frac{1}{5n} \left(1 - \frac{1}{5n}\right)^x, \quad x = 0, 1, 2, ...; \ n \in \mathbb{N}. \] Define \( Z_n = \frac{X_n}{n} \), \( n \in \mathbb{N} \), and let \( V \) be a random variable. If \( Z_n \xrightarrow{d} V \) as \( n \to \infty \), then which of the following statements is/are correct?

A

\(V \) has normal distribution with mean 5 and variance 25

B

\(V \) has chi-square distribution with 5 degrees of freedom

C

\(V \) has exponential distribution with mean 5

D

\(e^{-\frac{V}{5}} \) has uniform distribution over \((0, 1) \)

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