List of top Statistics Questions asked in IIT JAM MS

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables having U(0, 1) distribution. Let Y_n = n min{X₁, X₂ , … , X_n}, n ≥ 1. If Y_n converges to Y in distribution, then the median of Y equals __________ (round off to 2 decimal places)

Consider the linear regression model y_i = β₀ + β₁x_i + ∈i , i = 1, 2, … , 6, where β₀ and β₁ are unknown parameters and ∈_i ’s are independent and identically distributed random variables having N(0, 1) distribution. The data on (x_i, y_i) are given in the following table.

xi	1.0	2.0	2.5	3.0	3.5	4.5
yi	2.0	3.0	3.5	4.2	5.0	5.4

If \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are the least squares estimates of β₀ and β₁, respectively, based on the above data, then \(\hat{β}0 + \hat{β}1\) equals __________ (round off to 2 decimal places)

Let X₁, X₂ , … , X₉ be a random sample from a N(μ, σ2) distribution, where μ ∈ \(\R\) and σ > 0 are unknown. Let the observed values of \(\overline{X}=\frac{1}{9}\sum^9_{i=1}X_i\) and \(S^2=\frac{1}{8}\sum^9_{i=1}(X_i-\overline{X})^2\) be 9.8 and 1.44, respectively. If the likelihood ratio test is used to test the hypothesis H₀ : μ = 8.8 against H₁ : μ > 8.8, then the p-value of the test equals __________ (round off to 3 decimal places)

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} 8xy, & 0 < y < x < 1, \\ 0, & \text{otherwise}. \end{cases} \] Then \( 9 \, \text{Cov}(X, Y) \) equals

The area bounded between two parabolas \( y = x^2 + 4 \) and \( y = -x^2 + 6 \) is

Let \( X_1, X_2, X_3, X_4, X_5 \) be independent random variables with \[ X_1 \sim N(200, 8), \, X_2 \sim N(104, 8), \, X_3 \sim N(108, 15), \, X_4 \sim N(120, 15), \, X_5 \sim N(210, 15). \] Let \[ U = \frac{X_1 + X_2}{2}, \quad V = \frac{X_3 + X_4 + X_5}{3}. \] Then \( P(U>V) \) equals

If \( Y = \log_{10} X \) has \( N(\mu, \sigma^2) \) distribution with moment generating function \( M_Y(t) = e^{\mu t + \frac{\sigma^2 t^2}{2}} \), then \( P(X<1000) \) equals

Let \( X_1, X_2, X_3 \) be independent random variables with the common probability density function
\[ f(x) = \begin{cases} 2e^{-2x}, & \text{if} \, x > 0, \\ 0, & \text{otherwise}. \end{cases} \] Let \( Y = \min \{ X_1, X_2, X_3 \}, \, E(Y) = \mu_y \) and \( \text{Var}(Y) = \sigma_y^2 \). Then \( P(Y > \mu_y + \sigma_y) \) equals

The volume of the region \[ R = \left\{ (x, y, z) \in \mathbb{R}^3 : x + y + z \leq 3, y^2 \leq 4x, 0 \leq x \leq 1, y \geq 0, z \geq 0 \right\} \] is

A tangent is drawn on the curve \( y = \frac{1}{3} \sqrt{x^3} \), \( x>0 \) at the point \( P \left( 1, \frac{1}{3} \right) \), which meets the x-axis at \( Q \). Then the length of the closed curve \( OQPO \), where \( O \) is the origin, is

Let \( X \) be a continuous random variable with the probability density function
\[ f(x) = \begin{cases} \frac{x}{8}, & \text{if} \, 0 < x < 2, \\ k, & \text{if} \, 2 \leq x \leq 4, \\ \frac{6 - x}{8}, & \text{if} \, 4 < x < 6, \\ 0, & \text{otherwise}. \end{cases} \] Then \( P(1 < X < 5) \) equals

Let \( X \) and \( Y \) be discrete random variables with \( P(Y \in \{0, 1\}) = 1 \), \[ P(X = 0) = \frac{3}{4}, \, P(X = 1) = \frac{1}{4}, \, P(Y = 1 | X = 1) = \frac{3}{4}, \, P(Y = 0 | X = 0) = \frac{7}{8}. \] Then \( 3P(Y = 1) - P(Y = 0) \) equals

Let \( X \) be a random variable with the probability density function \[ f(x | r, \lambda) = \frac{x^{r-1} e^{-x/\lambda}}{\lambda^r (r-1)!}, \quad x>0, \, \lambda>0, \, r>0. \] If \( E(X) = 2 \) and \( \text{Var}(X) = 2 \), then \( P(X<1) \) equals

Let a unit vector \( \mathbf{v} = (v_1, v_2, v_3)^T \) be such that \( A\mathbf{v} = 0 \), where
\[ A = \begin{pmatrix} \frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix}. \] Then the value of \( \sqrt{6} \left( |v_1| + |v_2| + |v_3| \right) \) equals

Let \( u_n = \frac{18n + 3}{(3n - 1)^2 (3n + 2)^2} \), \( n \in \mathbb{N} \). Then \[ \sum_{n=1}^{\infty} u_n \quad \text{equals} \]

Let \( X_1, X_2, \dots, X_{100} \) be i.i.d. random variables with \( E(X_1) = 0, E(X_1^2) = \sigma^2 \), where \( \sigma>0 \). Let \[ S = \sum_{i=1}^{100} X_i. \, \text{If an approximate value of} \, P(S \leq 30) \, \text{is} \, 0.9332, \, \text{then} \, \sigma^2 \, \text{equals}. \]

Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x,y) = \begin{cases} \frac{1}{2} e^{-x}, & \text{if} \, |y| \leq x, \, x > 0, \\ 0, & \text{otherwise}. \end{cases} \] Then \( E(X | Y = -1) \) equals

Let \[ F(x) = \int_0^x e^{t^2 - 3t - 5} \, dt, \quad x>0. \] Then the number of roots of \( F(x) = 0 \) in the interval \( (0, 4) \) is

For \( j = 1, 2, \dots, 5 \), let \( P_j \) be the matrix of order \( 5 \times 5 \) obtained by replacing the \( j^{th} \) column of the identity matrix of order \( 5 \times 5 \) with the column vector \( v = (5, 4, 3, 2, 1)^T \). Then the determinant of the matrix product \( P_1 P_2 P_3 P_4 P_5 \) is

Let the random variable \( X \) have uniform distribution on the interval \( (0, 1) \) and \( Y = -2 \log X \). Then \( E(Y) \) equals

Let \( X_1, X_2, X_3, X_4 \) be a random sample from an \( N(\theta, 1) \) distribution, where \( \theta \in (-\infty, \infty) \). Suppose the null hypothesis \( H_0: \theta = 1 \) is to be tested against the hypothesis \( H_1: \theta<1 \) at \( \alpha = 0.05 \) level of significance. For what observed values of \( \sum_{i=1}^4 X_i \), the uniformly most powerful test would reject \( H_0 \)?

Let \( X \) and \( Y \) be discrete random variables with the joint probability mass function \[ p(x, y) = \frac{1}{25} (x^2 + y^2), \quad \text{if} \, x = 1, 2; \, y = 0, 1, 2. \] Then \( P(Y = 1 | X = 1) \) equals

Let \( X_1, X_2, X_3, X_4 \) be a random sample from an \( N(\mu, \sigma^2) \) distribution. Let \( \bar{X} = \frac{1}{4} \sum_{i=1}^4 X_i \) and \[ \tilde{Y} = \frac{15}{7} \sum_{i=1}^4 X_i. \] If \( \bar{X} \) has a t-distribution, then \( (\nu - k) \) equals

Let \( f : [0, \frac{\pi}{2}] \to \mathbb{R} \) be defined as \[ f(x) = ax + \beta \sin x, \] where \( a, \beta \in \mathbb{R} \). Let \( f \) have a local minimum at \( x = \frac{\pi}{4} \) with \[ f' \left( \frac{\pi}{4} \right) = -\frac{4}{\sqrt{2}}. \] Then \( 8\sqrt{2a} + 4 \beta \) equals

Let \( X_1, X_2, \dots, X_n \) (where \( n \geq 2 \)) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{x}{\theta^2} e^{-x/\theta}, & x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \). Which of the following statistics is (are) sufficient but NOT complete?