List of top Statistics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

Let X₁ , X₂ , … , X_n(n ≥ 2) be a random sample from Exp(\(\frac{1}{\theta}\)) distribution, where θ > 0 is unknown. If \(\overline{X}=\frac{1}{n}\sum^n_{i=1}X_i\), then which one of the following statements is NOT true ?

Let X₁,X₂ , … , X_n(n ≥ 3) be a random sample from a N(μ,σ²) distribution, where μ ∈ \(\R\) and σ > 0 are both unknown. Then which one of the following is a simple null hypothesis ?

Let X₍₁₎ < X₍₂₎ < ⋯ < X₍₉₎ be the order statistics corresponding to a random sample of size 9 from U(0, 1) distribution. Then which one of the following statements is NOT true ?

Let 2.5, -1.0, 0.5, 1.5 be the observed values of a random sample of size 4 from a continuous distribution with the probability density function
\(f(x)=\frac{1}{8}e^{-|x-2|}+\frac{3}{4\sqrt{2\pi}}e^{-\frac{1}{2}(x-\theta)^2},\ \ x \in \R,\)
where θ ∈ \(\R\) is unknown. Then the method of moments estimate of θ equals __________ (round off to 2 decimal places)

Let Ω = {1, 2, 3, … } be the sample space of a random experiment and suppose that all subsets of Ω are events. Further, let P be a probability function such that P({i}) > 0 for all i ∈ Ω. Then which of the following statements is/are true ?

A year is chosen at random from the set of years {2012, 2013, … , 2021}. From the chosen year, a month is chosen at random and from the chosen month, a day is chosen at random. Given that the chosen day is the 29^th of a month, the conditional probability that the chosen month is February equals

In a store, the daily demand for milk (in litres) is a random variable having Exp(λ) distribution, where λ > 0. At the beginning of the day, the store purchases c (> 0) litres of milk at a fixed price b (> 0) per litre. The milk is then sold to the customers at a fixed price s (> b) per litre. At the end of the day, the unsold milk is discarded. Then the value of c that maximizes the expected net profit for the store equals

Let 0.2, 1.2, 1.4, 0.3, 0.9, 0.7 be the observed values of a random sample of size 6 from a continuous distribution with the probability density function
\(f(x) = \begin{cases}    1, & 0< x \le \frac{1}{2} \\     \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise,}\end{cases}\)
where θ > \(\frac{1}{2}\) is unknown. Then the maximum likelihood estimate and the method of moments estimate of θ, respectively, are

Let X₁, X₂, X₃, X₄ be a random sample from a continuous distribution with the probability density function f(x) = \(\frac{1}{2}e^{-|x-\theta|}\), x ∈ \(\R\), where θ ∈ \(\R\) is unknown. Let the corresponding order statistics be denoted by X₍₁₎ < X₍₂₎ < X₍₃₎ < X₍₄₎. Then which of the following statements is/are true ?

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 (X, Y) be a random vector having the joint probability density function
\(f(x,y)=\begin{cases} \frac{\sqrt2}{\sqrt{\pi}}e^{-2x}e^{-\frac{(y-x)^2}{2}}, & 0 <x< ∞,-∞<y<∞\\ 0,& \text{otherwise} \end{cases}\)
Then E(Y) equals

Let X be a random variable having binomial distribution with parameters n(> 1) and p(0 < p < 1). Then \(E(\frac{1}{1+X})\) equals

Let X₁, X₂, X₃, X₄ be a random sample from a Poisson distribution with unknown mean λ > 0. For testing the hypothesis
H₀ : λ = 1 against H₁ : λ = 1.5,
let β denote the power of the test that rejects H₀ if and only if \(∑_{i=1}^4 𝑋_𝑖  ≥ 5\). Then which one of the following statements is true ?

Let X₁, X₂, X₃, X₄ be a random sample from a distribution with the probability mass function
\(f(x) = \begin{cases}    \theta^x(1-\theta)^{1-x}, & x=0,1 \\     0, & \text{otherwise}, \end{cases}\)
where θ ∈ (0, 1) is unknown. Let 0 < α ≤ 1. To test the hypothesis \(H_0:\theta=\frac{1}{2}\) against \(H_1:\theta>\frac{1}{2},\), consider the size α test that rejects H₀ if and only if \(\sum^4_{i=1} 𝑋𝑖 ≥ k_α\), for some 𝑘_α ∈ {0, 1, 2, 3, 4}. Then for which one of the following values of α, the size α test does NOT exist ?

Let X₁, X₂ , … , X_n (n > 1) be a random sample from a N(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Let 0 < α < 1. To test the hypothesis H₀ : μ = 0 against H₁ : μ = δ, where δ > 0 is a constant, let β denote the power of the size α test that rejects H₀ if and only if \(\frac{1}{n}\sum^n_{i=1}X_i > c_{\alpha}\) , for some constant c_α. Then which of the following statements is/are true ?

Let X be a random variable with the moment generating function
\(M(t)=\frac{1}{(1-4t)^5},t \lt\frac{1}{4}.\)
Then the lower bounds for P(X < 40), using Chebyshev’s inequality and Markov’s inequality, respectively, are

Let X₁, X₂ ,…, X₂₅ be a random sample from a 𝑁(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Consider testing of the hypothesis H₀ : μ = 5.2 against H₁ : μ = 5.6. The null hypothesis is rejected if and only if \(\frac{1}{25}\sum^{25}_{i=1}X_i > k\) , for some constant k. If the size of the test is 0.05, then the probability of type-II error equals __________ (round off to 2 decimal places)

Five men go to a restaurant together and each of them orders a dish that is different from the dishes ordered by the other members of the group. However, the waiter serves the dishes randomly. Then the probability that exactly one of them gets the dish he ordered equals __________ (round off to 2 decimal places)

Let X₁ and X₂ be two independent and identically distributed discrete random variables having the probability mass function
\(f(x) = \begin{cases} (\frac{1}{2})^x, & x=1,2,3,... \\   0, & \text{otherwise. }  \end{cases}\)
Then P(min{X₁ , X₂} ≥ 5) equals

Let (X, Y) be a random vector having bivariate normal distribution with parameters E(X) = 0, Var(X) = 1, E(Y) = −1, Var(Y) = 4 and ρ(X, Y) = \(-\frac{1}{2}\) , where ρ(X, Y) denotes the correlation coefficient between X and Y. Then P(X + Y > 1 | 2X − Y = 1) equals

A vaccine, when it is administered to an individual, produces no side effects with probability \(\frac{4}{5}\), mild side effects with probability \(\frac{2}{15}\) and severe side effects with probability \(\frac{1}{15}\). Assume that the development of side effects is independent across individuals. The vaccine was administered to 1000 randomly selected individuals. If X₁ denotes the number of individuals who developed mild side effects and X₂ denotes the number of individuals who developed severe side effects, then the coefficient of variation of X₁ + X₂ equals __________ (round off to 2 decimal places)

Let A and B be two events such that 0 < P(A) < 1 and 0 < P(B) < 1. Then which one of the following statements is NOT true ?

Suppose that four persons enter a lift on the ground floor of a building. There are seven floors above the ground floor and each person independently chooses her exit floor as one of these seven floors. If each of them chooses the topmost floor with probability \(\frac{1}{3}\) and each of the remaining floors with an equal probability, then the probability that no two of them exit at the same floor equals

Let X and Y be two independent and identically distributed random variables having U(0, 1) distribution. Then P(X² < Y < X) equals __________ (round off to 2 decimal places)

Let X₁ ,X₂ , … , X₁₆ be a random sample from a N(4μ, 1) distribution and Y₁ ,Y₂ , … , Y₈ be a random sample from a N(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Assume that the two random samples are independent. If you are looking for a confidence interval for μ based on the statistic \(8\overline{X} + \overline{Y}\), where \(\overline{X}=\frac{1}{16}\sum^{16}_{i=1}X_i\) and \(\overline{Y}=\frac{1}{8}\sum^8_{i=1}Y_i\), then which one of the following statements is true ?