Random Variables, Moments and Transformations - Probability - Statistics Previous Year Questions

1

2019 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2019

A fair die is rolled two times independently. Given that the outcome on the first roll is 1, the expected value of the sum of the two outcomes is

A

4

B

4.5

C

3

D

5.5

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2

2019 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2019

For \(k = 1, 2, ..., 10\), let the probability density function of the random variable \(X_k\) be \[ f_{X_k}(x) = \begin{cases} \frac{e^{-\frac{x}{k}}}{k}, & x > 0 \\ 0, & \text{otherwise.} \end{cases} \] Then \(E(\sum_{k=1}^{10} k X_k)\) is equal to ....

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3

2019 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2019

Let \( X \) be a random variable with uniform distribution on the interval \( (-1,1) \) and \( Y = (X+1)^2 \). Then the probability density function \( f(y) \) of \( Y \), over the interval \( (0,4) \), is

A

\(\frac{3\sqrt{Y}}{16} \)

B

\(\frac{1}{4\sqrt{y}} \)

C

\(\frac{1}{6\sqrt{y}} \)

D

\(\frac{1}{\sqrt{y}} \)

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4

2019 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2019

Let \(X\) be a random variable with characteristic function \(\phi_X(\cdot)\) such that \(\phi_X(2\pi) = 1\). Let \(\mathbb{Z}\) denote the set of integers. Then \(P(X \in \mathbb{Z})\) is equal to ...

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5

2020 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2020

The moment generating function of a random variable \(X\) is given by \(M_X(t) = \frac{1}{6} + \frac{1}{3}e^t + \frac{1}{3}e^{2t} + \frac{1}{6}e^{3t}, \quad -\infty < t < \infty\). Then \(P(X \le 2)\) equals

A

\(\frac{1}{3}\)

B

\(\frac{1}{6}\)

C

\(\frac{1}{2}\)

D

\(\frac{5}{6}\)

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6

2020 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2020

The characteristic function of a random variable X is given by
\[ \phi_X(t) = \begin{cases} \frac{\sin t \cos t}{t}, & \text{for } t \neq 0 \\ 1, & \text{for } t = 0 \end{cases} \]
Then \( P\left( |X| \leq \frac{2}{\pi} \right) = \) ___________ (correct up to two decimal places).

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7

2020 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2020

The total number of standard \( 4 \times 4 \) Latin squares is ___________

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8

2021 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2021

Let $X$ be a random variable having distribution function \(F(x) = \begin{cases} 0, & x < 1, \\ \frac{a}{2}, & 1 \le x < 2, \\ \frac{c}{6}, & 2 \le x < 3, \\ 1, & x \ge 3, \end{cases}\) where $a$ and $c$ are appropriate constants. Let $A_n = \left[ 1 + \frac{1}{n}, 3 - \frac{1}{n} \right], n \ge 1$, and $A = \cup_{i=1}^{\infty} A_i$. If $P(X \le 1) = \frac{1}{2}$ and $E(X) = \frac{5}{3}$, then $P(X \in A)$ equals ________ (round off to 2 decimal places).

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9

2021 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2021

Let \(X\) be a random variable having the probability density function \[f(x) = \begin{cases} \frac{3}{13}(1-x)(9-x), & 0 < x < 1, \\ 0, & \text{otherwise}. \end{cases}\] Then \(\frac{4}{3} E[X(X^2 - 15X + 27)]\) equals __________ (round off to 2 decimal places).

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10

2021 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2021

Let \(X\) be a random variable having the moment generating function
\(M(t) = \frac{e^t - 1}{t(1 - t)}, \quad t < 1.\).
Then \(P(X > 1)\) equals __________ (round off to 2 decimal places).

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11

2021 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2021

Let \(\{X_n\}_{n \geq 1}\) be a sequence of independent and identically distributed random variables each having uniform distribution on \([0, 3]\). Let \(Y\) be a random variable, independent of \(\{X_n\}_{n \geq 1}\), having probability mass function
\(P(Y = k) = \begin{cases} \frac{1}{(e - 1)k!}, & k = 1, 2, ..., \\ 0, & \text{otherwise}. \end{cases}\).
Then \(P(\max\{X_1, X_2, ..., X_Y\} \leq 1)\) equals __________ (round off to 2 decimal places).

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12

2022 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2022

Let \(X_i\), \(i = 1, 2, ..., n\), be i.i.d. random variables from a normal distribution with mean 1 and variance 4. Let \(S_n = X_1^2 + X_2^2 + \cdots + X_n^2\). If \(Var(S_n)\) denotes the variance of \(S_n\), then the value of \(\lim_{n \to \infty} \left( \frac{Var(S_n)}{n} - \left( \frac{E(S_n)}{n} \right)^2 \right)\) (in integer) is equal to ______

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13

2022 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2022

Let \(X\) be a random variable with the probability density function \(f(x) = \begin{cases} c(x - [x]), & 0 < x < 3, \\ 0, & \text{elsewhere}, \end{cases}\) where \(c\) is a constant and \([x]\) denotes the greatest integer less than or equal to \(x\). If \(A = \left[ \frac{1}{2}, 2 \right]\), then \(P(X \in A)\) (rounded off to two decimal places) is equal to ______

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14

2022 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2022

Let \(X\) and \(Y\) be two random variables such that the moment generating function of \(X\) is \(M(t)\) and the moment generating function of \(Y\) is \(H(t) = \left( \frac{3}{4} e^{2t} + \frac{1}{4} \right) M(t)\), where \(t \in (-h, h)\), \(h > 0\). If the mean and the variance of \(X\) are \(\frac{1}{2}\) and \(\frac{1}{4}\), respectively, then the variance of \(Y\) (in integer) is equal to ______

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15

2022 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2022

Consider the following statements:
(I) Let a random variable \( X \) have the probability density function
$$f_X(x) = \frac{1}{2} e^{-|x|}, \quad -\infty < x < \infty.$$
Then there exist i.i.d. random variables \( X_1 \) and \( X_2 \) such that \( X \) and \( X_1 - X_2 \) have the same distribution.
(II) Let a random variable \( Y \) have the probability density function
$$f_Y(y) = \begin{cases} \frac{1}{4}, & -2 < y < 2, \\ 0, & \text{elsewhere}. \end{cases}$$
Then there exist i.i.d. random variables \( Y_1 \) and \( Y_2 \) such that \( Y \) and \( Y_1 - Y_2 \) have the same distribution.
Then which of the above statements is/are true?

A

(I) only

B

(II) only

C

Both (I) and (II)

D

Neither (I) nor (II)

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16

2022 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2022

Let \( X \) be a random variable such that
\( P\left(\frac{a}{2\pi} X \in \mathbb{Z}\right) = 1, \; a > 0, \)
where \( \mathbb{Z} \) denotes the set of all integers. If \( \phi_X(t), t \in \mathbb{R}, \) denotes the characteristic function of \( X, \) then which of the following is/are true?

A

\(\phi_X(a) = 1 \)

B

\(\phi_X(\cdot) \) is periodic with period \(a \)

C

\(|\phi_X(t)| < 1 \) for all \(t \neq a \)

D

\(\int_0^{2\pi} e^{-itn} \phi_X(t) dt = \pi P\left(X = \frac{2\pi n}{a}\right), n \in \mathbb{Z}, \; i = \sqrt{-1} \)

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17

2023 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2023

Let \( \Phi(\cdot) \) denote the cumulative distribution function of a standard normal random variable. If the random variable \( X \) has the cumulative distribution function \[ F(x) = \begin{cases} \Phi(x) & \text{if } x < -1 \\ \Phi(x+1) & \text{if } x \ge -1, \end{cases} \] then which one of the following statements is true?

A

\(P(X \le -1) = \frac{1}{2} \)

B

\(P(X = -1) = \frac{1}{2} \)

C

\(P(X < -1) = \frac{1}{2} \)

D

\(P(X \le 0) = \frac{1}{2} \)

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18

2023 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2023

Let \( X \) be a random variable having Poisson distribution with mean \( \lambda > 0 \). Then \( E\left(\frac{1}{X+1} \mid X > 0\right) \) equals

A

\(\frac{1 - e^{-\lambda} - \lambda e^{-\lambda}}{\lambda(1 - e^{-\lambda})} \)

B

\(\frac{1 - e^{-\lambda}}{\lambda} \)

C

\(\frac{1 - e^{-\lambda} - \lambda e^{-\lambda}}{\lambda} \)

D

\(\frac{1 - e^{-\lambda}}{\lambda + 1} \)

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19

2023 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2023

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 \( Y = \log_e X \), then \( P(Y < 1 \mid Y < 2) \) equals

A

\(\frac{e}{1+e} \)

B

\(\frac{e-1}{e+1} \)

C

\(\frac{1}{1+e} \)

D

\(\frac{1}{e-1} \)

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20

2023 · Statistics · Probability · Random Variables, Moments and Transformations

Statistics (ST) 2023

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?

A

\(\lim_{n \to \infty} P\left(-\frac{1}{2} + \frac{1}{n} < X < -\frac{1}{n}\right) = \frac{5}{8} \)

B

\(\lim_{n \to \infty} P\left(-\frac{1}{2} - \frac{1}{n} < X < \frac{1}{n}\right) = \frac{5}{8} \)

C

\(\lim_{n \to \infty} P\left(X = \frac{1}{n}\right) = \frac{1}{2} \)

D

\(P(X = 0) = \frac{1}{3} \)

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