List of top Questions asked in GATE Statistics

Three villages P, Q, and R are located in such a way that the distance PQ = 13 km, QR = 14 km, and RP = 15 km, as shown in the figure. A straight road joins Q and R. It is proposed to connect P to this road QR by constructing another road. What is the minimum possible length (in km) of this connecting road?
Note: The figure shown is representative.

For the clock shown in the figure, if
O = O Q S Z P R T, and
X = X Z P W Y O Q,
then which one among the given options is most appropriate for P?

“His life was divided between the books, his friends, and long walks. A solitary man, he worked at all hours without much method, and probably courted his fatal illness in this way. To his own name there is not much to show; but such was his liberality that he was continually helping others, and fruits of his erudition are widely scattered, and have gone to increase many a comparative stranger’s reputation.” (From E.V. Lucas’s “A Funeral”)
Based only on the information provided in the above passage, which one of the following statements is true?
 

A bag contains Violet (V), Yellow (Y), Red (R), and Green (G) balls. On counting them, the following results are obtained:
(i) The sum of Yellow balls and twice the number of Violet balls is 50.
(ii) The sum of Violet and Green balls is 50.
(iii) The sum of Yellow and Red balls is 50.
(iv) The sum of Violet and twice the number of Red balls is 50.
Which one of the following Pie charts correctly represents the balls in the bag?
 

In the context of the given figure, which one of the following options correctly represents the entries in the blocks labelled (i), (ii), (iii), and (iv), respectively?

An electricity utility company charges ₹7 per kWh. If a 40-watt desk light is left on for 10 hours each night for 180 days, what would be the cost of energy consumption? If the desk light is on for 2 more hours each night for the 180 days, what would be the percentage-increase in the cost of energy consumption?
 

Let \( (X, Y)^T \) follow a bivariate normal distribution with \[ E(X) = 2, \, E(Y) = 3, \, {Var}(X) = 16, \, {Var}(Y) = 25, \, {Cov}(X, Y) = 14. \] Then \[ 2\pi \left( \Pr(X>2, Y>3) - \frac{1}{4} \right) \] equals _________ (rounded off to two decimal places).

Let \( X_1, X_2 \) be a random sample from a population having probability density function
\[ f_{\theta}(x) = \begin{cases} e^{(x-\theta)} & \text{if } -\infty < x \leq \theta, \\ 0 & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is an unknown parameter. Consider testing \( H_0: \theta \geq 0 \) against \( H_1: \theta < 0 \) at level \( \alpha = 0.09 \). Let \( \beta(\theta) \) denote the power function of a uniformly most powerful test. Then \( \beta(\log_e 0.36) \) equals ________ (rounded off to two decimal places).

Let \( X_1, X_2, \dots, X_7 \) be a random sample from a population having the probability density function \[ f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x}, \quad x>0, \] where \( \lambda>0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = \alpha \lambda \) be the corresponding bias, where \( \alpha \) is a real constant. Then the value of \( \frac{1}{\alpha} \) equals __________ (answer in integer).

Let \( \{ X_n \}_{n \geq 1} \) be a sequence of independent random variables with \[ \Pr(X_n = -\frac{1}{2^n}) = \Pr(X_n = \frac{1}{2^n}) = \frac{1}{2}, \quad \forall n \in \mathbb{N}. \] Suppose that \( \sum_{i=1}^{n} X_i \) converges to \( U \) as \( n \to \infty \). Then \( 6 \Pr(U \leq \frac{2}{3}) \) equals ___________ (answer in integer).

The service times (in minutes) at two petrol pumps \( P_1 \) and \( P_2 \) follow distributions with probability density functions \[ f_1(x) = \lambda e^{-\lambda x}, \quad x>0 \quad {and} \quad f_2(x) = \lambda^2 x e^{-\lambda x}, \quad x>0, \] respectively, where \( \lambda>0 \). For service, a customer chooses \( P_1 \) or \( P_2 \) randomly with equal probability. Suppose, the probability that the service time for the customer is more than one minute, is \( 2e^{-2} \). Then the value of \( \lambda \) equals _________ (answer in integer).

The moment generating functions of three independent random variables \( X, Y, Z \)  are respectively given as: \[ M_X(t) = \frac{1}{9}(2 + e^t)^2, \quad t \in \mathbb{R}, \] \[ M_Y(t) = e^{e^t - 1}, \quad t \in \mathbb{R}, \] \[ M_Z(t) = e^{2(e^t - 1)}, \quad t \in \mathbb{R}. \] Then \( 10 \cdot \Pr(X>Y + Z) \) equals __________ (rounded off to two decimal places).

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be defined as \[ f(x, y) = x^2 y^2 + 8x - 4y. \] The number of saddle points of \( f \) is _________ (answer in integer).

For \( Y \in \mathbb{R}^n \), \( X \in \mathbb{R}^{n \times p} \), and \( \beta \in \mathbb{R}^p \), consider a regression model \[ Y = X \beta + \epsilon, \] where \( \epsilon \) has an \( n \)-dimensional multivariate normal distribution with zero mean vector and identity covariance matrix. Let \( I_p \) denote the identity matrix of order \( p \). For \( \lambda>0 \), let \[ \hat{\beta}_n = (X^T X + \lambda I_p)^{-1} X^T Y, \] be an estimator of \( \beta \). Then which of the following options is/are correct?

Let \( X = (X_1, X_2, X_3)^T \) be a 3-dimensional random vector having multivariate normal distribution with mean vector \( (0, 0, 0)^T \) and covariance matrix
\[ \Sigma = \begin{pmatrix} 4 & 0 & 0 <br> 0 & 9 & 0 <br> 0 & 0 & 4 \end{pmatrix}. \]
{Let } \( \alpha^T = (2, 0, -1) \) { and } \( \beta^T = (1, 1, 1) \). 
Then which of the following statements is/are correct?