List of top Statistics Questions asked in GATE ST

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by

\[ f(x) = \begin{cases} x^3 \sin x, & \text{if } x = 0 \text{ or } x \text{ is irrational}, \\ \frac{1}{q^3}, & \text{if } x = \frac{p}{q},\; p \in \mathbb{Z} \setminus \{0\},\; q \in \mathbb{N},\; \text{and } \gcd(p,q) = 1, \end{cases} \]

where \( \mathbb{R} \) denotes the set of all real numbers, \( \mathbb{Z} \) denotes the set of all integers, \( \mathbb{N} \) denotes the set of all positive integers, and \( \gcd(p,q) \) denotes the greatest common divisor of \( p \) and \( q \). Then which one of the following statements is true?

Let \( X_1, X_2, X_3 \) be three uncorrelated random variables with common variance \( \sigma^2<\infty \). Let \( Y_1 = 2X_1 + X_2 + X_3 \), \( Y_2 = X_1 + 2X_2 + X_3 \), and \( Y_3 = X_1 + X_2 + 2X_3 \). Then which of the following statements is/are true?

Let \( \Omega = \{1, 2, 3, \dots \} \) represent the collection of all possible outcomes of a random experiment with probabilities \( P(\{n\}) = a_n \) for \( n \in \Omega \). Then which one of the following statements is NOT true?

For \( \alpha>0 \), let \[ \{ X^{(\alpha)}_n \}_{n \geq 1} \text{ be a sequence of independent random variables such that} \quad P(X^{(\alpha)}_n = 1) = \frac{1}{n^{2\alpha}} = 1 - P(X^{(\alpha)}_n = 0). \] Let \( S = \{ \alpha>0 : X^{(\alpha)}_n \text{ converges to } 0 \text{ almost surely as } n \to \infty \}. \) \text{Then the infimum of \( S \) equals} _________ \text{ (round off to 2 decimal places).}

Let \( A \) and \( B \) be two events such that \( P(B) = \frac{3}{4} \) and \( P(A \cup B^C) = \frac{1}{2} \). If \( A \) and \( B \) are independent, then \( P(A) \) equals _________ (round off to 2 decimal places).

A fair die is rolled twice independently. Let \( X \) and \( Y \) denote the outcomes of the first and second roll, respectively. Then \[ E(X + Y \mid (X - Y)^2 = 1) \] The value of \( E(X + Y \mid (X - Y)^2 = 1) \) is _________ (round off to 2 decimal places).

Evaluate the limit: \[ \lim_{n \to \infty} \frac{(2^n + n 2^n \sin^2 \frac{n}{2})}{(2n - n \cos \frac{1}{n})} \] The value of the limit is _________ (round off to 2 decimal places).

Let \[ I = 4 \int_0^{\frac{1}{\sqrt{2}}} \int_0^x \frac{1}{\sqrt{x^2 + y^2}} \, dy \, dx \] Then the value of \( e^{l+\pi} \) is _________ (round off to 2 decimal places).

Let \( X \) be a non-constant positive random variable such that \( E(X) = 9 \). Then which one of the following statements is true?