List of top Questions asked in GATE Statistics

Let $ \{X_n\}_{n \geq 1} $ be a Markov chain with state space $ \{1, 2, 3\} $ and transition probability matrix \[ P = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] Then $ P(X_2 = 1 \mid X_1 = 1, X_3 = 2) $ (rounded off to two decimal places) equals:

Suppose that $ (X_1, X_2, X_3) $ has a $ N_3(\mu, \Sigma) $ distribution with \[ \mu = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \quad \text{and} \quad \Sigma = \begin{pmatrix} 2 & 2 & 1 \\ 2 & 5 & 1 \\ 1 & 1 & 1 \end{pmatrix}. \] Given that $ \Phi(-0.5) = 0.3085 $, where $ \Phi(.) $ denotes the cumulative distribution function of a standard normal random variable, \[ P\left( (X_1 - 2X_2 + 2X_3)^2 < \frac{7}{2} \right) \text{ (rounded off to two decimal places) equals ...............} \]

Let $ X $ be a random variable with probability density function \[ f(x) = \begin{cases} \frac{1}{x^2} & \text{if } x \geq 1, \\ 0 & \text{otherwise.} \end{cases} \] If $ Y = \log X $, then $ P(Y < 1 \mid Y < 2) $ equals

Let $ \{N(t)\}_{t \geq 0} $ be a Poisson process with rate 1. Consider the following statements.
[(I)] $ P(N(3) = 3 \mid N(5) = 5) = \binom{5}{3} \left(\frac{3}{5}\right)^3 \left(\frac{2}{5}\right)^2 $.
[(II)] If $ S_5 $ denotes the time of occurrence of the 5th event for the above Poisson process, then $ E(S_5 \mid N(5) = 3) = 7 $.
Which of the above statements is/are true?

Let $ X_1, X_2, \dots, X_n $ be a random sample of size $ n $ from a population having probability density function \[ f(x; \mu) = \begin{cases} e^{-(x-\mu)} & \text{if } \mu \leq x < \infty \\ 0 & \text{otherwise} \end{cases} \] where $ \mu \in \mathbb{R} $ is an unknown parameter. If $ \hat{M} $ is the maximum likelihood estimator of the median of $ X_1 $, then which one of the following statements is true?

Suppose that $ (X, Y) $ has joint probability mass function \[ P(X = 0, Y = 0) = P(X = 1, Y = 1) = \theta, \quad P(X = 1, Y = 0) = P(X = 0, Y = 1) = \frac{1}{2} - \theta, \] where $ 0 \leq \theta \leq \frac{1}{2} $ is an unknown parameter. Consider testing $ H_0 : \theta = \frac{1}{4} $ against $ H_1 : \theta = \frac{1}{3} $, based on a random sample \[ \{ (X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n) \} \] from the above probability mass function. Let $ M $ be the cardinality of the set \[ \{ i : X_i = Y_i, 1 \leq i \leq n \}, \] If $ m $ is the observed value of $ M $, then which one of the following statements is true?

Let $ g(x) = f(x) + f(2 - x) $ for all $ x \in [0, 2] $, where $ f : [0, 2] \to \mathbb{R} $ is continuous on $ [0, 2] $ and twice differentiable on $ (0, 2) $. If $ g' $ denotes the first derivative of $ g $ and $ f'' $ denotes the second derivative of $ f $, then which of the following statements is NOT true?

Let $ X $ be a random variable with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < -1, \\ \frac{1}{4}(x + 1) & \text{if } -1 \leq x < 0, \\ \frac{1}{4}(x + 3) & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] Which one of the following statements is true?

Let $ (X, Y) $ have joint probability mass function \[ p(x, y) = \begin{cases} \frac{c}{2x + y + 2} & \text{if } x = 0, 1, 2, \dots; \, y = 0, 1, 2, \dots; \, x \neq y, \\ 0 & \text{otherwise}. \end{cases} \] Then which one of the following statements is true?

Let $ X_1, X_2, \dots, X_{10} $ be a random sample of size 10 from a $ N_3(\mu, \Sigma) $ distribution, where $ \mu $ and non-singular $ \Sigma $ are unknown parameters. If \[ \overline{X_1} = \frac{1}{5} \sum_{i=1}^{5} X_i, \quad \overline{X_2} = \frac{1}{5} \sum_{i=6}^{10} X_i, \] \[ S_1 = \frac{1}{4} \sum_{i=1}^{5} (X_i - \overline{X_1})(X_i - \overline{X_1})^T, \quad S_2 = \frac{1}{4} \sum_{i=6}^{10} (X_i - \overline{X_2})(X_i - \overline{X_2})^T, \] then which one of the following statements is NOT true?

Let $ A $ be a $ 3 \times 3 $ real matrix such that \[ A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 4 \end{bmatrix}. \] Then which of the following statements is/are true?

Let $ X $ be a random variable with probability density function \[ f(x) = \begin{cases} e^{-x} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases} \] For $ a < b $, if $ U(a, b) $ denotes the uniform distribution over the interval $ (a, b) $, then which of the following statements is/are true?

Suppose that X is a discrete random variable with the following probability mass function: \[ P(X = 0) = \frac{1}{2}(1 + e^{-1}), P(X = k) = \frac{e^{-1}}{2 k!} for k = 1, 2, 3, .... \] Which of the following statements is/are true?

Suppose that $ U $ and $ V $ are two independent and identically distributed random variables each having probability density function \[ f(x) = \begin{cases} \lambda^2 x e^{-\lambda x} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \] where $ \lambda > 0 $. Which of the following statements is/are true?

Let $ (X, Y) $ have joint probability mass function \[ p(x, y) = \begin{cases} \frac{e^{-2}}{x! (y - x)!} & \text{if } x = 0, 1, 2, \dots, y; \, y = 0, 1, 2, \dots \\ 0 & \text{otherwise}. \end{cases} \] Then which of the following statements is/are true?

Let $ \{X_n\}_{n \geq 1} $ be a sequence of independent and identically distributed random variables with mean 0 and variance 1, all of them defined on the same probability space. For $ n = 1, 2, 3, \dots $, let \[ Y_n = \frac{1}{n} \left( X_1 X_2 + X_3 X_4 + \dots + X_{2n-1} X_{2n} \right). \] Then which one of the following statements is/are true?

Consider the orthonormal set \[ v_1 = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}, v_2 = \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \end{bmatrix}, v_3 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \end{bmatrix} \] with respect to the standard inner product on $ \mathbb{R}^3 $. If $ u = \begin{bmatrix} a \\ b \\ c \end{bmatrix} $ is the vector such that inner products of $ u $ with $ v_1, v_2 $ and $ v_3 $ are 1, 2 and 3, respectively, then $ a^2 + b^2 + c^2 $ (in integer) equals ................

Let X be a random sample of size 1 from a population with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < 0, \\ 1 - (1 - x)^\theta & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] where $ \theta > 0 $ is an unknown parameter. To test $ H_0 : \theta = 1 $ against $ H_1 : \theta = 2 $, consider using the critical region $ \{ x \in \mathbb{R} : x < 0.5 \} $. If $ \alpha $ and $ \beta $ denote the level and power of the test, respectively, then $ \alpha + \beta $ (rounded off to two decimal places) equals:

Let $ \{0.13, 0.12, 0.78, 0.51\} $ be a realization of a random sample of size 4 from a population with cumulative distribution function $ F(.) $. Consider testing $ H_0: F = F_0 $ against $ H_1: F \neq F_0 $, where \[ F_0(x) = \begin{cases} 0 & \text{if } x < 0, \\ x & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1. \end{cases} \] Let $ D $ denote the Kolmogorov-Smirnov test statistic. If $ P(D > 0.669) = 0.01 $ under $ H_0 $ and $ \psi = \begin{cases} 1 & \text{if } H_0 \text{ is accepted at level 0.01}, \\ 0 & \text{otherwise} \end{cases} $
then based on the given data, the observed value of } $ D + \psi $
(rounded off to two decimal places) equals ...............

Consider the probability space $(\Omega, \mathcal{G}, P)$, where $\Omega = [0,2]$ and $\mathcal{G} = \{\emptyset, \Omega, [0,1], (1,2]\}$. Let $X$ and $Y$ be two functions on $\Omega$ defined as} \[ X(\omega) = \begin{cases} 1 & \text{if } \omega \in [0,1] \\ 2 & \text{if } \omega \in (1,2] \end{cases} \] and \[ Y(\omega) = \begin{cases} 2 & \text{if } \omega \in [0,1.5] \\ 3 & \text{if } \omega \in (1.5,2] \end{cases} \] Then which one of the following statements is true?

Consider the following statements. (I) Let $ A $ and $ B $ be two $ n \times n $ real matrices. If $ B $ is invertible, then $ \text{rank}(BA) = \text{rank}(A) $. (II) Let $ A $ be an $ n \times n $ real matrix. If $ A^2x = b $ has a solution for every $ b \in \mathbb{R}^n $, then $ Ax = b $ also has a solution for every $ b \in \mathbb{R}^n $. Which of the above statements is/are true?

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown. The area of the shaded region is ............... cm$^2$.

Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), $P(x)$, of a variable $x$?

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. The figure below consists of 20 unit squares arranged as shown. In addition to the given black squares, up to 5 more may be coloured black. Which one among the following options depicts the minimum number of boxes that must be coloured black to achieve two lines of symmetry? (The figure is representative)