List of top Questions asked in IIT JAM MS- 2021

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: