Laws of Large Numbers and Central Limit Theorem - Convergence of Random Variables - Statistics Previous Year Questions

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

2019 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2019

Let \(\{X_k\}_{k \geq 1}\) be a sequence of independent and identically distributed Bernoulli random variables with success probability \(p \in (0,1)\). Then, as \(n \to \infty\), \(\frac{1}{n} \sum_{k=1}^n (X_k)^k\) converges almost surely to

A

\(p\)

B

\(\frac{1}{1-p}\)

C

\(\frac{1-p}{p}\)

D

\(1\)

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3

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 random variables with mean \(\theta\) and variance \(\theta\), where \(\theta > 0\). Then \(\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^2}\) is a consistent estimator of

A

\(\frac{1}{1+\theta}\)

B

\(\frac{1+\theta}{\theta}\)

C

\(\frac{1}{\theta}\)

D

\(\frac{\theta}{1+\theta}\)

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4

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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5

2021 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2021

Let \(\{X_n\}_{n \ge 1}\) be a sequence of independent and identically distributed random variables each having uniform distribution on \([0,2]\). For \(n \ge 1\), let \[Z_n = -\log_e \left( \prod_{i=1}^n (2 - X_i) \right)^{\frac{1}{n}}.\] Then, as \(n \to \infty\), the sequence \(\{Z_n\}_{n \ge 1}\) converges almost surely to __________ (round off to 2 decimal places).

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6

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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7

2023 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2023

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having mean 4 and variance 9. If \( Y_n = \frac{1}{n} \sum_{i=1}^n X_i \) for \( n \geq 1 \), then \( \lim_{n \to \infty} E \left[ \left( \frac{Y_n - 4}{\sqrt{n}} \right)^2 \right] \) (in integer) equals ______________

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8

2023 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2023

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having probability density function \( f(x) = \begin{cases} e^{-x} & \text{if } x > 0 \\ 0 & \text{otherwise}. \end{cases} \) For \( n \geq 1 \), let \( Y_n = |X_{2n} - X_{2n-1}| \). If \( \bar{Y}_n = \frac{1}{n} \sum_{i=1}^n Y_i \) for \( n \geq 1 \) and \( \{\sqrt{n} (e^{-\bar{Y}_n} - e^{-1})\}_{n \geq 1} \) converges in distribution to a normal random variable with mean \( 0 \) and variance \( \sigma^2 \), then \( \sigma^2 \) (rounded off to two decimal places) equals __________

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9

2024 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2024

Let \(\{X_n\}_{n \geq 1}\) be a sequence of independent and identically distributed random variables having probability density function \(f(x) = \begin{cases} e^{-(x-\theta)} & \text{if } x \geq \theta \\ 0 & \text{otherwise} \end{cases}\) where \(\theta > 0\). Consider the following statements: (I) \(\frac{1}{n} \sum_{i=1}^n X_i\) converges in probability to \(\frac{\theta + 1}{2}\) as \(n \to \infty\). (II) \(\lim_{n \to \infty} E(\min(X_1, X_2, ..., X_n)) = \theta\). Which of the above statements is/are true?

A

Only (I)

B

Only (II)

C

Both (I) and (II)

D

Neither (I) nor (II)

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10

2025 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2025

Let \(\{X_n\}_{n \geq 1}\) be a sequence of i.i.d. random variables with the common probability density function \(f(x) = \frac{1}{\pi(1+x^2)},\ -\infty < x < \infty\). Define \(Y_n = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(X_n)\) for \(n = 1, 2, ...\). Then which one of the following options is correct?

A

\(\frac{1}{n} \sum_{i=1}^n Y_i \xrightarrow{P} \frac{1}{2}\) as \(n \to \infty\)

B

\(\frac{1}{n} \sum_{i=1}^n Y_i \xrightarrow{P} 0\) as \(n \to \infty\)

C

\(\frac{1}{n} \sum_{i=1}^n X_i \xrightarrow{P} 0\) as \(n \to \infty\)

D

\(\frac{1}{n} \sum_{i=1}^n X_i \xrightarrow{P} \frac{1}{2}\) as \(n \to \infty\)

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11

2025 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2025

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent random variables with
\( \Pr\left(X_n = -\frac{1}{2^n}\right) = \Pr\left(X_n = \frac{1}{2^n}\right) = \frac{1}{2} \quad \forall n \in \mathbb{N} \).
Suppose that \( \sum_{i=1}^n X_i \xrightarrow{d} U \) as \( n \to \infty \). Then \( 6 \Pr\left(U \leq \frac{2}{3}\right) \) equals ______________ (answer in integer).

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12

2026 · Statistics · Convergence of Random Variables · Laws of Large Numbers and Central Limit Theorem

Statistics (ST) 2026

Let $\{X_k\}_{k\geq 1}$ be a sequence of independent random variables such that $X_{2k-1} \sim \text{Bin}(1,\theta)$, and $X_{2k} \sim \text{Bin}(1,1-\theta)$, $k=1,2,3,\ldots$, where $\theta \in (0,1)$. Let $\{Y_k\}_{k\geq 1}$ be another sequence of independent and identically distributed random variables such that $Y_k \sim \text{Poisson}(\lambda), \lambda>0$. Define, for $n \in \mathbb{N}$, $S_{2n} = \sum_{k=1}^{n} (X_{2k-1} - X_{2k} + 1 - 2\theta)$, $W_n = \sum_{k=1}^{n} Y_k^2$ and $\sigma_{2n}^2 = 2n\theta(1-\theta)$. Then which of the following statements is/are correct?

A

$\frac{n S_{2n}}{\sigma_{2n} W_n} \xrightarrow{d} N(0,1)$, as $n \to \infty$

B

$\frac{n S_{2n}}{\sigma_{2n} W_n} \xrightarrow{d} N\left(0, \frac{1}{\lambda^2(1+\lambda)^2}\right)$, as $n \to \infty$

C

$\frac{n S_{2n} + \sigma_{2n} W_n}{n \sigma_{2n}} \xrightarrow{d} N(\lambda(1+\lambda), 1)$, as $n \to \infty$

D

$\frac{n S_{2n} + \sigma_{2n} W_n}{n \sigma_{2n}} \xrightarrow{d} N\left(0, \frac{1}{\lambda^2(1+\lambda)^2}\right)$, as $n \to \infty$

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