List of top Questions asked in GATE Data Science and Artificial Intelligence

Suppose \( X \) and \( Y \) are random variables. The conditional expectation of \( X \) given \( Y \) is denoted by \( E[X | Y] \). Then \( E[E[X | Y]] \) equals:

If a real variable \(x\) satisfies \(3^{x^2} = 27 \times 9^x\), then the value of \(\frac{2^{x^2}}{(2^x)^2}\) is:

Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk.

A rectangle has a length \(L\) and a width \(W\), where \(L > W\). If the width, \(W\), is increased by 10%, which one of the following statements is correct for all values of \(L\) and \(W\)?
Select the most appropriate option to complete the above sentence.

A 4 × 4 digital image has pixel intensities (U) as shown in the figure. The number of pixels with \( U \leq 4 \) is:

We ____________ tennis in the lawn when it suddenly started to rain.
Select the most appropriate option to complete the above sentence.

Courage : Bravery :: Yearning :
Select the most appropriate option to complete the analogy.

A bag contains 5 white balls and 10 black balls. In a random experiment, \( n \) balls are drawn from the bag one at a time with replacement. Let \( S_n \) denote the total number of black balls drawn in the experiment. The expectation of \( S_{100} \) denoted by \( E[S_{100}] \) is \( \_\ \_\ \_\ \_\ \_\ \) (Round off to one decimal place).

Consider a two-class problem in \( \mathbb{R}^d \) with class labels red and green. Let \( \mu_{\text{red}} \) and \( \mu_{\text{green}} \) be the means of the two classes. Given test sample \( x \in \mathbb{R}^d \), a classifier calculates the squared Euclidean distance (denoted by \( \| \cdot \|^2 \)) between \( x \) and the means of the two classes and assigns the class label that the sample \( x \) is closest to. That is, the classifier computes \[ f(x) = \| \mu_{\text{red}} - x \|^2 - \| \mu_{\text{green}} - x \|^2 \] and assigns the label red to \( x \) if \( f(x)<0 \), and green otherwise. Which of the following statements is/are correct?

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( |f(x) - f(y)| \leq (x - y)^2 \) for all \( x, y \in \mathbb{R} \). Then \[ f(1) - f(0) = \ \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Let \( Y = Z^2 \), \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is a normal random variable with mean \( \mu \) and variance \( \sigma^2 \). The variance of \( Y \) is

Evaluate the following limit: \[ \lim_{t \to \infty} \sqrt{t^2 + t - t} \]

It is given that \( P(X \geq 2) = 0.25 \) for an exponentially distributed random variable \( X \) with \( E[X] = \frac{1}{\lambda} \), where \( E[X] \) denotes the expectation of \( X \). What is the value of \( \lambda \)?
(\(\ln\) denotes natural logarithm)

Let \( p \) and \( q \) be any two propositions. Consider the following propositional statements.
S1: \( p \rightarrow q \), S2: \( \neg p \land q \), S3: \( \neg p \lor q \), S4: \( \neg p \lor \neg q \)
where \( \land \) denotes conjunction (AND operation), \( \lor \) denotes disjunction (OR operation), and \( \neg \) denotes negation (NOT operation).
(Note: \( \equiv \) denotes logical equivalence)
Which one of the following options is correct?

Weight of a person can be expressed as a function of their age. This function usually varies from person to person. Suppose this function is identical for two brothers and is monotonically increasing until the age of 50 years, after which it monotonically decreases. Let \( a_1 \) and \( a_2 \) (in years) denote the ages of the brothers, where \( a_1 < a_2 \).

Which one of the following statements is correct about their ages on the day they attain the same weight?

Given the table facts: \[ \text{facts (D1, D2, val)} \] with the following data: