List of top Questions asked in GATE Statistics

Let \( X \) be a random variable having the Poisson distribution with mean \( \log_e 2 \). Then \( E\left( e^{(\log_e 3)X} \right) \) equals:
Let \( S = \{(x, y, z) \in \mathbb{R}^3 \setminus \{(0,0,0)\} : z = -(x + y)\} \). Denote:

\[ S^\perp = \{(p, q, r) \in \mathbb{R}^3 : px + qy + rz = 0 \text{ for all } (x, y, z) \in S\}. \]

Then which one of the following options is correct?
Among the following four statements about countability and uncountability of different sets, which is the correct statement?
Let \( f : [0, \infty) \to [0, \infty) \) be a differentiable function with \( f(x)>0 \) for all \( x>0 \), and \( f(0) = 0 \). Further, \( f \) satisfies} \[ (f(x))^2 = \int_{0}^{x} \left( (f(t))^2 + f(t) \right) \, dt, \, x>0. \] {Then which one of the following options is correct?

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.


 

A business person buys potatoes of two different varieties P and Q, mixes them in a certain ratio and sells them at ₹192 per kg.
The cost of the variety P is ₹800 for 5 kg.
The cost of the variety Q is ₹800 for 4 kg.
If the person gets 8% profit, what is the P : Q ratio (by weight)?

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?


 

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?
 

“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?
 

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?
 

The President, along with the Council of Ministers, ............. to visit India next week.
Select the most appropriate option to complete the above sentence.
Even though I had planned to go skiing with my friends, I had to ............. at the last moment because of an injury.
Select the most appropriate option to complete the above sentence.
Let \( X_1, \dots, X_n \) be a random sample from a uniform distribution over the interval \( \left( -\frac{\theta}{2}, \frac{\theta}{2} \right) \), where \( \theta>0 \) is an unknown parameter. Then which of the following options is/are correct?
Consider a five-digit number PQRST that has distinct digits P, Q, R, S, and T, and satisfies the following conditions:
1. \( P<Q \)
2. \( S>P>T \)
3. \( R<T \)
If integers 1 through 5 are used to construct such a number, the value of P is: