List of top Questions asked in IIT JAM MS

Let \( X \) be a continuous random variable having the moment generating function \[ M(t) = \frac{e^t - 1}{t}, \quad t \ne 0. \] Let \( \alpha = P(48X^2 - 40X + 3>0) \) and \( \beta = P((\ln X)^2 + 2\ln X - 3>0) \). Then, the value of \( \alpha - 2\ln \beta \) is equal to:
Let \( X_1, X_2, ..., X_n \) (\( n \ge 3 \)) be a random sample from \( \text{Poisson}(\theta) \), where \( \theta>0 \) is unknown, and let \( T = \sum_{i=1}^{n} X_i \). Then, the uniformly minimum variance unbiased estimator (UMVUE) of \( e^{-2\theta}\theta^3 \) is:
Let \( X_1, X_2, ..., X_n \) (\( n \ge 2 \)) be a random sample from \( U(\theta - 5, \theta + 5) \), where \( \theta \in (0, \infty) \) is unknown. Let \( T = \max(X_1, ..., X_n) \) and \( U = \min(X_1, ..., X_n) \). Then, which of the following statements is TRUE?
Let \( X \) and \( Y \) be random variables having chi-square distributions with 6 and 3 degrees of freedom respectively. Then, which of the following statements is TRUE?
If the series \(\sum_{n=1}^{\infty} a_n\) converges absolutely, then which of the following series diverges?
There are three urns labeled 1, 2, 3.
Urn 1: 2 white, 2 black; Urn 2: 1 white, 3 black; Urn 3: 3 white, 1 black.
Two coins are tossed independently, each with \(P(\text{head}) = 0.2\).
Urn 1 is selected if 2 heads occur, Urn 3 if 2 tails occur, otherwise Urn 2 is selected. A ball is drawn at random from the chosen urn. Find \[ P(\text{Urn 1 is selected} \mid \text{Ball drawn is white}) \]
Let \(X\) be a random variable with \[ f(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty \] Then, which of the following statements is FALSE?
Let \( \{X_n\}_{n \ge 1} \) be i.i.d. random variables distributed as \( N(0,1) \). Then find \[ \lim_{n \to \infty} P\left( \frac{\sum_{i=1}^{n} X_i^2 - 3n}{\sqrt{32n}} \le \sqrt{6} \right) \] is equal to:
Consider independent Bernoulli trials with success probability \( p = \frac{1}{3} \). The probability that three successes occur before four failures is:
Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables and \(Z = \left|\frac{X}{Y}\right|\). Then, which of the following expectations is finite?
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^7 + 5x^3 + 11x + 15 \). Then, which of the following statements is TRUE?
The value of the limit \[ \lim_{x \to 0} \frac{e^{-3x} - e^{x} + 4x}{5(1 - \cos x)} \] is equal to:
The value of the limit \[ \lim_{n \to \infty} \sum_{k=0}^{n} \binom{2n}{k} \frac{1}{4^n} \] is equal to:
Let \(X\) be a \(U(0,1)\) random variable and \(Y = X^2\). If \(\rho\) is the correlation coefficient between \(X\) and \(Y\), then \(48\rho^2\) is equal to:
The value of the limit \[ \lim_{n \to \infty} \left( \left(1 + \frac{1}{n}\right) \left(1 + \frac{2}{n}\right) \cdots \left(1 + \frac{n}{n}\right) \right)^{\frac{1}{n}} \] is equal to:

Let $X$ be a random variable having the Poisson(4) distribution and let $E$ be an event such that $P(E|X = i) = 1 - 2^{-i}$, $i = 0, 1, 2, \ldots$. Then $P(E)$ equals ........... (round off to two decimal places).
 

Let $X_1, X_2,$ and $X_3$ be independent random variables such that $X_1 \sim N(47, 10)$, $X_2 \sim N(55, 15)$, and $X_3 \sim N(60, 14)$. Then $P(X_1 + X_2 \ge 2X_3)$ equals ............ (round off to two decimal places).

Let $U \sim F_{5,8}$ and $V \sim F_{8,5}$. If $P[U > 3.69] = 0.05$, then the value of $c$ such that $P[V > c] = 0.95$ equals ................ (round off to two decimal places).
 

Let the sample mean based on a random sample from Exp($\lambda$) distribution be 3.7. Then the maximum likelihood estimate of $1 - e^{-\lambda}$ equals ........... (round off to two decimal places).
 

Let $X$ be a single observation drawn from $U(0, \theta)$ distribution, where $\theta \in \{1, 2\}$. To test $H_0: \theta = 1$ against $H_1: \theta = 2$, consider the test procedure that rejects $H_0$ if and only if $X > 0.75$. If the probabilities of Type-I and Type-II errors are $\alpha$ and $\beta$, respectively, then $\alpha + \beta$ equals ......... (round off to two decimal places).
 

Let $\alpha$ be the real number such that the coefficient of $x^{125}$ in Maclaurin's series of $(x + \alpha^3)^3 e^x$ is $\dfrac{28}{124!}$. Then $\alpha$ equals ..............
 

Consider the matrix $M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & 1 & 4 \end{bmatrix}$. Let $P$ be a nonsingular matrix such that $P^{-1}MP$ is a diagonal matrix. Then the trace of the matrix $P^{-1}M^3P$ equals ........... 

Let $P$ be a $3 \times 3$ matrix having characteristic roots $\lambda_1 = -\dfrac{2}{3}$, $\lambda_2 = 0$ and $\lambda_3 = 1$. Define $Q = 3P^3 - P^2 - P + I_3$ and $R = 3P^3 - 2P$. If $\alpha = \det(Q)$ and $\beta = \text{trace}(R)$, then $\alpha + \beta$ equals .......... (round off to two decimal places).

Let $X$ and $Y$ be independent random variables with respective moment generating functions $M_X(t) = \dfrac{(8 + e^t)^2}{81}$ and $M_Y(t) = \dfrac{(1 + 3e^t)^3}{64}$, $-\infty < t < \infty$. Then $P(X + Y = 1)$ equals .............. (round off to two decimal places).
 

A computer lab has two printers handling certain types of printing jobs. Printer-I and Printer-II handle 40% and 60% of the jobs, respectively. For a typical printing job, printing time (in minutes) of Printer-I follows $N(10, 4)$ and that of Printer-II follows $U(1, 21)$. If a randomly selected printing job is found to have been completed in less than 10 minutes, then the conditional probability that it was handled by the Printer-II equals .......... (round off to two decimal places).