List of top Statistics Questions asked in IIT JAM MS

If \[ \int_0^1 \int_0^{\sqrt{1 - (y - 1)^2}} f(x, y) \, dx \, dy \] equals \[ \int_0^1 \int_0^x f(x, y) \, dy \, dx, \] then \( \alpha(x) \) and \( \beta(x) \) are
Let \( f: [0, 1] \to \mathbb{R} \) be a function defined as
\[ f(t) = \begin{cases} t^3 \left( 1 + \frac{1}{5} \cos(\log(e^t)) \right), & \text{if } t \in (0,1] \\ 0, & \text{if } t = 0 \end{cases} \] Let \( F: [0, 1] \to \mathbb{R} \) be defined as
\[ F(x) = \int_0^x f(t) \, dt \] Then \( F''(0) \) equals
Consider the function \[ f(x, y) = x^3 - y^3 - 3x^2 + 3y^2 + 7, \, x, y \in \mathbb{R}. \] Then the local minimum (\( m \)) and the local maximum (\( M \)) of \( f \) are given by
Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} x + y, & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases} \] Then \( P(X + Y > \frac{1}{2}) \) equals
Let \( x_1 = 1.1, x_2 = 0.5, x_3 = 1.4, x_4 = 1.2 \) be the observed values of a random sample of size four from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} e^{-\theta x}, & \text{if } x \geq \theta \\ 0, & \text{otherwise}, \quad \theta \in (-\infty, \infty) \end{cases} \] Then the maximum likelihood estimate of \( \theta^2 \) is
Let \( X \) be a discrete random variable with the probability mass function
\[ p(x) = k(1 + |x|)^2, \quad x = -2, -1, 0, 1, 2, \] where \( k \) is a real constant. Then \( P(X = 0) \) equals
Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables having common probability density function
\[ f(x) = \begin{cases} x e^{-x}, & x \geq 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \), \( n = 1, 2, \dots \). Then \[ \lim_{n \to \infty} P(\bar{X}_n = 2) \] equals
Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x>0, \ \theta>0 \] Which of the following estimators of \( \theta \) has the smallest variance for all \( \theta>0 \)?
Player \( P_1 \) tosses 4 fair coins and player \( P_2 \) tosses a fair die independently of \( P_1 \). The probability that the number of heads observed is more than the number on the upper face of the die, equals
Let \( X_1 \) and \( X_2 \) be i.i.d. continuous random variables with the probability density function
\[ f(x) = \begin{cases} 6x(1 - x), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Using Chebyshev's inequality, the lower bound of \( P \left( |X_1 + X_2 - 1| \leq \frac{1}{2} \right) \) is
Let \( X \) be a random variable with the moment generating function
\[ M_X(t) = \frac{1}{216} \left( 5 + e^t \right)^3, \quad t \in \mathbb{R}. \] Then \( P(X>1) \) equals
Let \( u, v \in \mathbb{R}^4 \) be such that \( u = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} \) and \( v = \begin{pmatrix} 5 \\ 3 \\ 2 \\ 1 \end{pmatrix} \). Then the equation \( u^T x = v \) has
Let \( \{a_n\} \) be a sequence defined as follows:
\[ a_1 = 1 \quad \text{and} \quad a_{n+1} = \frac{7a_n + 11}{21}, \quad n \in \mathbb{N}. \] Which of the following statements is TRUE?