List of top Statistics Questions asked in IIT JAM MS

Let \(X_1, X_2, \ldots, X_{10}\) be a random sample from the \(N(3,4)\) distribution, and let \(Y_1, Y_2, \ldots, Y_{15}\) be a random sample from the \(N(-3,6)\) distribution. Assume that the two samples are drawn independently. Define:
\[ \bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i \]
\[ \bar{Y} = \frac{1}{15} \sum_{j=1}^{15} Y_j \]
\[ S = \sqrt{\frac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2} \]
Then the distribution of \( U = \frac{\sqrt{5}(\bar{X} + \bar{Y})}{S} \) is:

For \( n \in \mathbb{N} \), let \( X_n \) be a random variable having the \( \text{Bin}(n, \frac{1}{4}) \) distribution. Then
\[\lim_{n \to \infty} \left[ P(X_n \leq 2n - \frac{\sqrt{3n}}{8}) + P\left(\frac{n}{6} \leq X_n \leq \frac{n}{3}\right) \right]\]
is equal to (You may use \( \Phi(0.5) = 0.6915 \), \( \Phi(1) = 0.8413 \), \( \Phi(1.5) = 0.9332 \), \( \Phi(2) = 0.9772 \)).

Let \( (X_1, Y_1), (X_2, Y_2), \ldots, (X_{20}, Y_{20}) \) be a random sample from the \( N_2(0, 0, 1, 1, \frac{3}{4}) \) distribution. Define
\[\bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad \bar{Y} = \frac{1}{20} \sum_{i=1}^{20} Y_i.\]
Then \( \text{Var}(\bar{X} - \bar{Y}) \) is equal to:

Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:

Suppose that the random variable \( X \) has an \( \text{Exp}\left(\frac{1}{5}\right) \) distribution, and for any \( x > 0 \), the conditional distribution of the random variable \( Y \), given \( X = x \), is \( N(x, 2) \). Then \( \text{Var}(X + Y) \) is equal to:

Let \( X \) be a random variable such that \( X \) and \( -X \) have the same distribution. Let \( Y = X^2 \) be a continuous random variable with the probability density function
\[g(y) = \begin{cases} \frac{e^{-y/2}}{\sqrt{2\pi y}}, & \text{if } y > 0 \\0, & \text{if } y \leq 0 \end{cases}.\]
Then \( E((X - 1)^4) \) is equal to:

Let \( f_1(x) \) be the probability density function of the \( N(0,1) \) distribution, and \( f_2(x) \) be the probability density function of the \( N(0, 6) \) distribution. Let \( Y \) be a random variable with probability density function
\[f(x) = 0.6 f_1(x) + 0.4 f_2(x), \quad -\infty < x < \infty.\]
Then \( \text{Var}(Y) \) is equal to:

Two fair coins \( S_1 \) and \( S_2 \) are tossed independently once. Let the events \( E, F, \) and \( G \) be defined as follows:
\( E \): Head appears on \( S_1 \)
\( F \): Head appears on \( S_2 \)  
\( G \): The same outcome (head or tail) appears on both \( S_1 \) and \( S_2 \)  
Then which of the following statements is NOT correct?

Let \( x_1, x_2, x_3, x_4 \) be the observed values from a random sample drawn from a \( N(\mu, \sigma^2) \) distribution, where \( \mu \in \mathbb{R} \) and \( \sigma \in (0, \infty) \) are unknown parameters. Let \( \bar{x} \) and \( s = \sqrt{\frac{1}{3} \sum_{i=1}^{4} (x_i - \bar{x})^2} \) be the observed be the observed sample mean sample standard deviation,repectively. For testing the hypotheses \( H_0: \mu = 0 \) against \( H_1: \mu \neq 0 \), the likelihood ratio test of size \( \alpha = 0.05 \) rejects \( H_0 \) if and only if \[\frac{|\bar{x}|}{s} > k.\] Then the value of \( k \) is given by:

Let \( X_1, X_2, \ldots, X_{20} \) be a random sample from the \( N(5, 2) \) distribution, and let \( Y_i = X_{2i} - X_{2i-1} \) for \( i = 1, 2, \ldots, 10 \). Then \( W = \frac{1}{4} \sum_{i=1}^{10} Y_i^2 \) has the distribution:

For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:

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

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 𝑋₁, 𝑋₂, be a sequence of 𝑖.𝑖.𝑑. random variables each having 𝑈(0,1) distribution. Let 𝑌 be a random variable having the distribution function 𝐺. Suppose that
\(lim𝑛_{n→∞} \frac{( 𝑋-1+𝑋_2+𝑋_𝑛}{4} ≤ 𝑥)=𝐺(𝑥)\), for all 𝑥 ∈ ℝ . 
Then, 𝑉𝑎𝑟(𝑌) equals

Let 𝑋₁, 𝑋₂, 𝑋₃ be a random sample from an 𝑁(𝜃, 1) distribution, where 𝜃 ∈ ℝ is an unknown parameter. Then, which one of the following conditional expectations does NOT depend on 𝜃 ?

A box contains a certain number of balls out of which 80% are white, 15% are blue and 5% are red. All the balls of the same color are indistinguishable. Among all the white balls, 𝛼% are marked defective, among all the blue balls, 6% are marked defective and among all the red balls, 9% are marked defective. A ball is chosen at random from the box. If the conditional probability that the chosen ball is white, given that it is defective, is 0.4, then 𝛼 equals ________

Let 𝑋 be a continuous random variable such that 𝑃(𝑋 ≥ 0)=1 and 𝑉𝑎𝑟(𝑋)<∞ . Then, 𝐸(𝑋 ²) is

Let 𝑋 and 𝑌 be jointly distributed random variables such that, for every fixed 𝜆>0, the conditional distribution of 𝑋 given 𝑌=𝜆 is the Poisson distribution with mean 𝜆. If the distribution of 𝑌 is 𝐺𝑎𝑚𝑚𝑎 (2, \(\frac{1}{2}\) ), then the value of 𝑃(𝑋= 0)+𝑃(𝑋=1) equals

Let 𝑋₁~Gamma(1,4), 𝑋₂~Gamma(2, 2) and 𝑋₃~Gamma(3, 4) be three independent random variables. If 𝑌=𝑋₁+2𝑋₂+𝑋₃, then 𝐸((\(\frac{y}{4}\))⁴ ) equals ______