Random Variables and Distributions - Probability and Statistics - Data Science & Artificial Intelligence Previous Year Questions

1

2024 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2024

Consider the following statements:
(i) The mean and variance of a Poisson random variable are equal.
(ii) For a standard normal random variable, the mean is zero and the variance is one.
Which ONE of the following options is correct?

A

Both (i) and (ii) are true

B

(i) is true and (ii) is false

C

(ii) is true and (i) is false

D

Both (i) and (ii) are false

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2

2024 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2024

A fair six-sided die (with faces numbered 1, 2, 3, 4, 5, 6) is repeatedly thrown independently.
What is the expected number of times the die is thrown until two consecutive throws of even numbers are seen?

A

2

B

4

C

6

D

8

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3

2024 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2024

Let \(X\) be a random variable uniformly distributed in the interval \([1, 3]\) and \(Y\) be a random variable uniformly distributed in the interval \([2, 4]\). If \(X\) and \(Y\) are independent of each other, the probability \(P(X \geq Y)\) is ______. (rounded off to three decimal places).

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4

2024 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2024

Let \(X\) be a random variable exponentially distributed with parameter \(\lambda > 0\). The probability density function of \(X\) is given by:
\[ f_X(x) = \begin{cases} \lambda e^{-\lambda x}, & x \geq 0 \\ 0, & \text{otherwise} \end{cases} \]
If \(5E(X) = Var(X)\), where \(E(X)\) and \(Var(X)\) indicate the expectation and variance of \(X\), respectively, the value of \(\lambda\) is ______. (rounded off to one decimal place).

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5

2024 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2024

Consider a joint probability density function of two random variables \(X\) and \(Y\):
\[ f_{X,Y}(x, y) = \begin{cases} 2xy, & 0 < x < 2,\ 0 < y < x \\ 0, & \text{otherwise} \end{cases} \]
Then, \(E[Y|X = 1.5]\) is ______.

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6

2024 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2024

Two fair coins are tossed independently. X is a random variable that takes a value of 1 if both tosses are heads and 0 otherwise. Y is a random variable that takes a value of 1 if at least one of the tosses is heads and 0 otherwise.
The value of the covariance of X and Y is ____ (rounded off to three decimal places).

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7

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

Let \(X\) be a continuous random variable whose cumulative distribution function (CDF) \(F_X(x)\), for some \(t\), is given as follows: \[ F_X(x) = \begin{cases} 0 & x \le t \\ \frac{x - t}{4 - t} & t \le x \le 4 \\ 1 & x \ge 4 \end{cases} \] If the median of \(X\) is 3, then what is the value of \(t\)?

A

2

B

1

C

-1

D

0

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8

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

Let \(X = aZ + b\), where \(Z\) is a standard normal random variable, and \(a, b\) are two unknown constants. It is given that \[ E[X] = 1, \quad E[(X - E[X])Z] = -2, \quad E[(X - E[X])^2] = 4, \] where \(E[X]\) denotes the expectation of random variable \(X\). The values of \(a, b\) are:

A

\(a = -2, b = 1\)

B

\(a = 2, b = -1\)

C

\(a = -2, b = -1\)

D

\(a = 1, b = 1\)

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9

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

It is given that \(P(X \ge 2) = 0.25\) 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)

A

\(\ln 2\)

B

\(\ln 4\)

C

\(\ln 3\)

D

\(\ln 0.25\)

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10

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

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

A

1

B

2

C

3

D

4

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11

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

Consider the cumulative distribution function (CDF) of a random variable \(X\):
\[F_X(x) = \begin{cases} 0 & x \le -1 \\ \frac{1}{4}(x+1)^2 & -1 \le x \le 1 \\ 1 & x \ge 1 \end{cases}\]
The value of \(P(X^2 \le 0.25)\) is

A

0.625

B

0.25

C

0.5

D

0.5625

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12

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

For \( x \in \mathbb{R} \), the floor function is denoted by \( f(x) = \lfloor x \rfloor \) and defined as follows \[ \lfloor x \rfloor = k, \quad k \le x < k+1, \] where \( k \) is an integer. Let \( Y = \lfloor X \rfloor \), where \( X \) is an exponentially distributed random variable with mean \( \frac{1}{\ln 10} \), where \( \ln \) denotes natural logarithm. For any positive integer \( \ell \), one can write the probability of the event \( Y = \ell \) as follows \[ P(Y = \ell) = q^{\ell}(1 - q) \] The value of \( q \) is

A

0.1

B

0.01

C

0.5

D

0.434

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13

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

Consider a coin-toss experiment where the probability of head showing up is \(p\). In the \(i^{th}\) coin toss, let \(X_i = 1\) if head appears, and \(X_i = 0\) if tail appears. Consider \(\widehat{p} = \frac{1}{n} \sum_{i=1}^n X_i\) where \(n\) is the total number of independent coin tosses. Which of the following statements is/are correct?

A

\(E[\widehat{p}] = p\)

B

\(E[\widehat{p}] = \frac{p}{n}\)

C

As \(n\) increases, variance of \(\widehat{p}\) decreases

D

Variance of \(\widehat{p}\) does not depend on \(n\)

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14

2025 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science & Artificial Intelligence (DA) 2025

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 \(E[S_{100}] = \)__________ (Round off to one decimal place)

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15

2026 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science and Artificial Intelligence (DA) 2026

Suppose a random variable $Z$ follows $Normal(\mu = 0, \sigma^2 = 1)$ distribution with probability density function $g(z)$ and cumulative distribution function $G(z)$. Another random variable $Y$ follows $t_1$ distribution with probability density function $h(y)$ and cumulative distribution function $H(y)$. Let $c$ be the positive real number for which $g(c) = h(c)$. Which of the following statements is/are correct?

A

$G(0) = H(0)$

B

$G(c) < H(c)$

C

$G(-c) < H(-c)$

D

$g(0) = h(0)$

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16

2026 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science and Artificial Intelligence (DA) 2026

Let \( X \) be an exponentially distributed random variable with mean \( \lambda (> 0) \). If \( P(X > 5) = 0.35 \), then the conditional probability \( P(X > 10 | X > 5) \) is __________. (Rounded off to two decimal places)

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17

2026 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science and Artificial Intelligence (DA) 2026

The value of \( \sum_{i=0}^{\infty} \sum_{j=1}^{\infty} 2^{-i} 3^{-j} \) is __________. (Answer in integer)

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18

2026 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science and Artificial Intelligence (DA) 2026

Let \(X\) and \(Y\) be two independent random variables. \(X\) follows \(Bernoulli(p = 0.3)\) distribution and \(Y\) follows \(Normal(\mu = 0, \sigma^2 = 100)\) distribution.
Which of the following options is the variance of \((2X - 1)Y\)?

A

100

B

90

C

49

D

21

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19

2026 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science and Artificial Intelligence (DA) 2026

Let \( L = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{e^{-n} n^k}{k!} \)
Which of the following is the value of \(L\)?

A

0.5

B

1.0

C

0

D

\(e^{-1}\)

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20

2026 · Data Science & Artificial Intelligence · Probability and Statistics · Random Variables and Distributions

Data Science and Artificial Intelligence (DA) 2026

Let \( X_1, X_2, ..., X_n \) be \( n \) independent random variables. Each of the random variables follows \( Normal(\mu = 0, \sigma^2 = 1) \) distribution. Define \( \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \).
Which of the following statements is/are correct?

A

\(\sum_{i=1}^n X_i^2 \) follows Chi-square distribution with \(n \) degrees of freedom.

B

\(\sum_{i=1}^n (X_i - \bar{X})^2 \) follows Chi-square distribution with \((n - 1) \) degrees of freedom.

C

\(X_1^2 + X_n^2 \) follows exponential distribution with mean 2.

D

\((\sqrt{n} \bar{X})^2 \) follows Chi-square distribution with 2 degrees of freedom.

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