List of top Questions asked in IIT JAM MS- 2020

Let $X_1, X_2, \ldots, X_n$ be a random sample from $N(\mu_1, \sigma^2)$ distribution and $Y_1, Y_2, \ldots, Y_m$ be a random sample from $N(\mu_2, \sigma^2)$ distribution, where $\mu_i \in \mathbb{R}, i = 1, 2$ and $\sigma > 0$. Suppose that the two random samples are independent. Define \[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \text{and} W = \frac{\sqrt{mn} (\bar{X} - \mu_1)}{\sqrt{\sum_{i=1}^{m} (Y_i - \mu_2)^2}}. \] Then which one of the following statements is TRUE for all positive integers $m$ and $n$? 
 

Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from $N(0,1)$ distribution and let \[ W = \frac{X_1^2}{X_1^2 + X_2^2 + X_3^2 + X_4^2 + X_5^2}. \] Then $E(W)$ equals

Let the joint probability density function of $(X, Y)$ be \[ f(x, y) = \begin{cases} 2e^{-(x + y)}, & 0 < x < y < \infty, \\ 0, & \text{otherwise.} \end{cases} \] 

Then $P\left(X < \dfrac{Y}{2}\right)$ equals 
 

Consider the simple linear regression model $y_i = \alpha + \beta x_i + \varepsilon_i$, where $\varepsilon_i$ are i.i.d. random variables with mean 0 and variance $\sigma^2$. Given that $n = 20$, $\sum x_i = 100$, $\sum y_i = 50$, $\sum x_i^2 = 600$, $\sum y_i^2 = 500$, and $\sum x_i y_i = 400$, find the least squares estimates of $\alpha$ and $\beta$.
Let $X_1, X_2, X_3, X_4$ be i.i.d. random variables having a continuous distribution. Then $P(X_3 < X_2 < \max(X_1, X_4))$ equals
If the volume of the region bounded by the paraboloid $z = x^2 + y^2$ and the plane $z = 2y$ is given by \[ \int_{0}^{\alpha} \int_{\beta(y)}^{2y} \int_{-\sqrt{z - y^2}}^{\sqrt{z - y^2}} dx \, dz \, dy \] then

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be defined by \[ f(x,y) = \begin{cases} \dfrac{2x^3 + 3y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \] 

Let $f_x(0,0)$ and $f_y(0,0)$ denote first order partial derivatives of $f(x,y)$ at $(0,0)$. Which one of the following statements is TRUE? 
 

Let $\alpha$ and $\beta$ be real numbers. If \[ \lim_{x \to 0} \frac{\tan 2x - 2 \sin \alpha x}{x(1 - \cos 2x)} = \beta, \] then $\alpha + \beta$ equals

Let $X_1, X_2, \dots, X_{10}$ be a random sample from $N(1,2)$ distribution. If $\bar{X} = \dfrac{1}{10} \sum_{i=1}^{10} X_i$ and $S^2 = \dfrac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2$, then $\text{Var}(S^2)$ equals
 

The area of the region bounded by the curves $y_1(x) = x^4 - 2x^2$ and $y_2(x) = 2x^2$, $x \in \mathbb{R}$, is
 

Let $f : [-1,3] \to \mathbb{R}$ be a continuous function such that $f$ is differentiable on $(-1,3)$, $|f'(x)| \le \dfrac{3}{2}$ for all $x \in (-1,3)$, $f(-1) = 1$ and $f(3) = 7$. Then $f(1)$ equals .................
 

Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?

Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{5}$. Then which of the following statements is/are TRUE?
 

For real constants $a$ and $b$, let \[ M = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ a & b \end{bmatrix} \] be an orthogonal matrix. Then which of the following statements is/are always TRUE?

Let the sequence $\{x_n\}_{n \ge 1}$ be given by $x_n = \sin \dfrac{n\pi}{6}$, $n = 1, 2, \ldots$. Then which of the following statements is/are TRUE?
 

Let $X_1, X_2, \ldots, X_n$ be a random sample from a distribution with probability density function \[ f_{\theta}(x) = \begin{cases} \theta (1 - x)^{\theta - 1}, & 0 < x < 1, \\ 0, & \text{otherwise}, \end{cases} \theta > 0. \] To test $H_0: \theta = 1$ against $H_1: \theta > 1$, the uniformly most powerful (UMP) test of size $\alpha$ would reject $H_0$ if 
 

Let $X_1, X_2, \ldots, X_n$ be a random sample from $U(\theta - 0.5, \theta + 0.5)$ distribution, where $\theta \in \mathbb{R}$. If $X_{(1)} = \min(X_1, X_2, \ldots, X_n)$ and $X_{(n)} = \max(X_1, X_2, \ldots, X_n)$, then which one of the following estimators is NOT a maximum likelihood estimator (MLE) of $\theta$?
 

Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{3}$. Let $X$ denote the number of trials required to get the second success. Then $P(X \ge 5)$ equals
 

Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always
 

Let $T: \mathbb{R}^3 \to \mathbb{R}^4$ be a linear transformation. If $T(1,1,0) = (2,0,0,0)$, $T(1,0,1) = (2,4,0,0)$, and $T(0,1,1) = (0,0,2,0)$, then $T(1,1,1)$ equals
 

The value of the integral \[ \int_{0}^{2} \int_{0}^{\sqrt{2x - x^2}} \sqrt{x^2 + y^2} \, dy \, dx \] is

Let $\{a_n\}_{n \ge 1}$ be a sequence of real numbers such that $a_1 = 1, a_2 = 7$, and $a_{n+1} = \dfrac{a_n + a_{n-1}}{2}$, $n \ge 2$. Assuming that $\lim_{n \to \infty} a_n$ exists, the value of $\lim_{n \to \infty} a_n$ is
 

Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables having $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$. Define \[ W = \frac{1}{2n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} (X_i - X_j)^2. \] Then $W$, as an estimator of $\sigma^2$, is 
 

Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables such that $E(X_i) = 1$ and $\text{Var}(X_i) = 1$. Then the approximate distribution of $\dfrac{1}{\sqrt{n}} \sum_{i=1}^n (X_{2i} - X_{2i-1})$, for large $n$, is
 

Consider the following system of linear equations: 
\[ \begin{cases} ax + 2y + z = 0 \\ y + 5z = 1  \\ by - 5z = -1 \end{cases} \] 

Which one of the following statements is TRUE?