List of top Questions asked in IIT JAM MS

Let \( X \) be a random variable having the Poisson distribution with mean \( 1 \). Let \( g: \mathbb{N} \cup \{0\} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} 1 & \text{if } x \in \{0, 2\} \\ 0 & \text{if } x \notin \{0, 2\} \end{cases}\]
Then \( E(g(X)) \) is equal to:

Let 𝑋 be a continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?

Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?

A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals

Let \( A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix} \) be a real matrix, where \( ad = 1 \) and \( c \ne 0 \). If \( A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \), then \( (\alpha, \beta, \gamma, \delta) \) is equal to

Let \( f(x,y) = x^2 + 3y^2 - \frac{2}{3} xy \) at \( (x,y) \in \mathbb{R}^2 \). Then

Let \( a_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \) and \( b_n = \sum_{k=1}^{n} \frac{1}{k^2} \) for all \( n \in \mathbb{N} \). Then

Which of the following functions represents a cumulative distribution function?

For \( n \geq 2 \), let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with \( E(X_1) = 0 \), \( \text{Var}(X_1) = 1 \), and \( E(X_1^4) < \infty \). Let \[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i \] and \[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2. \]
Then which of the following statements is/are always correct?

Let \( X \) and \( Y \) be continuous random variables having the joint probability density function
\[ f(x, y) = \begin{cases} e^{-x}, & \text{if } 0 \leq y < x < \infty \\0, & \text{otherwise}\end{cases} \]
Then which of the following statements is/are correct?

Let 𝑋~𝐹_6,2 and 𝑌~𝐹_2,6 . If 𝑃(𝑋 ≤ 2) =\( \frac{216}{343}\) and 𝑃 (𝑌≤\(\frac{1}{2}\) )=𝛼, then 686𝛼 equals

Let 𝐸₁, 𝐸₂ and 𝐸₃ be three events such that 𝑃(𝐸₁∩𝐸₂ )=\(\frac{1}{4}\) , 𝑃(𝐸₁ ∩ 𝐸₃ )=𝑃(𝐸₂ ∩ 𝐸₃ )=\(\frac{1}{5}\) and 𝑃(𝐸₁ ∩ 𝐸₂ ∩ 𝐸₃ ) =\(\frac{1}{6}\) . Then, among the events 𝐸₁ 𝐸₂ and 𝐸₃, the probability that at least two events occur, equals

Let 𝑃 be a 3×3 matrix having the eigenvalues 1,1 and 2. Let (1,−1,2) ^𝑇 be the only linearly independent eigenvector corresponding to the eigenvalue 1. If the adjoint of the matrix 2𝑃 is denoted by 𝑄, then which of the following statements is/are TRUE?

If, for some 𝛼∈(0, ∞),
\(∫^∞_02^{-x^2}dx=a\sqrt\pi,\) 
then 𝛼 equals (round off to two decimal places)

Let 𝑀 be a 3×3 real matrix. If 𝑃=𝑀+𝑀^𝑇 and 𝑄=𝑀-𝑀^𝑇, then which of the following statements is/are always TRUE?

Let 𝑋 be a random variable having the probability density function
\(f(x) =\begin{cases}   \frac{x}{2}       & \quad \text{if }0<x<2,\\  0, & \quad Otherwise \end{cases}\)
Then, 𝑉𝑎𝑟 (ln ( \(\frac{2}{x}\) )) equals ________

Let 𝑋 be a random variable having the probability density function
\(f(x) =\begin{cases} \frac{5}{x^6}, & \quad \text{if }x>1,\\ 0, & \quad Otherwise. \end{cases}\)
Then, which of the following statements is/are TRUE for the distribution of 𝑋 ?

Let 𝑋 be a random variable such that the moment generating function of 𝑋 exists in a neighborhood of zero and
\(E(X^n)=(-1)^n\frac{2}{5}+\frac{2^{n+1}}{5}+\frac{1}{5},\) n=1,2,3,…..
Then, 𝑃 (|𝑋-\(\frac{1}{2}\)|>1) equals

Let 𝑋 be a random variable having a probability mass function 𝑝(𝑥) which is positive only for non-negative integers. If
𝑝(𝑥+1)=(\(\frac{ ln\,\, 3}{𝑥+1}\)) 𝑝(𝑥), 𝑥=0, 1, 2, … , 
then 𝑉𝑎𝑟(𝑋) equals

Let 𝑋 be a random variable having a probability density function
\(f(x; θ) =\begin{cases}  (3-θ){x^2-θ}      & \quad \text{if }0<x<1,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃 ∈ {0, 1}. For testing the null hypothesis 𝐻₀ : 𝜃=0 against 𝐻₁ : 𝜃=1, the power of the most powerful test, at the level of significance 𝛼=0.125, equals

Let 𝑋₁~𝑁(2, 1), 𝑋₂~𝑁(−1,4) and 𝑋₃~𝑁(0, 1) be mutually independent random variables. Then, the probability that exactly two of these three random variables are less than 1, equals____(round off to two decimal places)

Let (𝑋, 𝑌) be a random vector having the joint probability density function

where 𝛼 is a positive constant. Then, 𝑃(𝑋 > 𝑌) equal

Among all the points on the sphere 𝑥²+𝑦²+𝑧²=24, the point (𝛼, 𝛽, 𝛾) is closest to the point (1, 2,−1). Then, the value of 𝛼+𝛽+𝛾 equals

Let 𝑥₁=2.1, 𝑥₂=4.2, 𝑥₃=5.8 and 𝑥₄=3.9 be the observed values of a random sample 𝑋₁, 𝑋₂, 𝑋₃ and 𝑋₄ from a population having a probability density function
\(f(x) =\begin{cases}   \frac{x}{θ^2}e^-\frac{x}{θ}     & \quad \text{if }x >0,\\  0, & \quad Otherwise \end{cases}\)
where 𝜃 ∈ (0, ∞) is an unknown parameter. Then, the maximum likelihood estimate of 𝑉𝑎𝑟(𝑋₁) equals ________

A point (𝑎, 𝑏) is chosen at random from the rectangular region [0, 2]×[0, 4]. Then, the probability that the area of the region
𝑅={(𝑥, 𝑦) ∈ ℝ × ℝ ∶ 𝑏𝑥 + 𝑎𝑦 ≤ 𝑎𝑏, 𝑥, 𝑦 ≥ 0} 
will be less than 2, equals