Joint, Marginal and Conditional Distributions - Joint Distributions - Statistics Previous Year Questions

1

2019 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2019

The probability density function of the random vector \((X, Y)\) is given by \[ f_{X,Y}(x, y) = \begin{cases} c, & 0 < x < y < 1 \\ 0, & \text{otherwise.} \end{cases} \] Then the value of c is equal to .....

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2

2019 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2019

Let \( (X_1, X_2) \) be a random vector following bivariate normal distribution with mean vector \( (0,0) \), Variance\( (X_1) = \) Variance\( (X_2) = 1 \) and correlation coefficient \( \rho \), where \( |\rho| < 1 \). Then \( P(X_1 + X_2 > 0) \) is equal to ...

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3

2019 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2019

Let \((X, Y)\) be a bivariate random vector with probability density function \[ f_{X,Y}(x,y) = \begin{cases} e^{-y}, & 0 < x < y, \\ 0, & otherwise. \end{cases} \] Then the regression of \(Y\) on \(X\) is given by

A

\(X + 1\)

B

\(\frac{X}{2}\)

C

\(\frac{Y}{2}\)

D

\(Y + 1\)

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4

2019 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2019

Consider the trinomial distribution with the probability mass function \( P(X=x, Y=y) = \frac{7!}{x!y!(7-x-y)!}(0.6)^x(0.2)^y(0.2)^{7-x-y} \), \( x \geq 0, y \geq 0 \), and \( x+y \leq 7 \). Then \( E(Y|X=3) \) is equal to ...

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5

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let the joint probability mass function of (X, Y, Z) be
\[ P(X = x, Y = y, Z = z) = \frac{10!}{x! y! z! k!} (0.2)^x (0.3)^y (0.4)^z (0.1)^k, \]
where \( k = 10 - x - y - z; \; x, y, z = 0, 1, ..., 10; \; x + y + z \leq 10 \).
Then the variance of the random variable \( Y + Z \) equals ___________ (correct up to one decimal place).

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6

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let the random vector \( \underline{X} = (X_1, X_2, X_3) \) have the joint probability density function \( f_{\underline{X}}(x_1, x_2, x_3) = \begin{cases} \frac{1 - \sin x_1 \sin x_2 \sin x_3}{8 \pi^3}, & 0 \le x_1, x_2, x_3 \le 2 \pi \\ 0, & \text{otherwise} \end{cases} \). Which of the following statements is TRUE?

A

\(X_1, X_2 \) and \(X_3 \) are mutually independent

B

\(X_1, X_2 \) and \(X_3 \) are pairwise independent

C

\((X_1, X_2) \) and \(X_3 \) are independently distributed

D

Variance of \(X_1 + X_2 \) is \(\pi^2 \)

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7

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let the joint distribution of \( (X, Y) \) be bivariate normal with mean vector \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \) and variance-covariance matrix \( \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix} \), where \( -1 < \rho < 1 \). Then \( E[\max(X, Y)] \) equals

A

\(\sqrt{\frac{1 - \rho}{\pi}} \)

B

\(\sqrt{\frac{1 - \rho}{\pi}} \)

C

0

D

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

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8

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let $(X, Y)$ be a random vector such that, for any $y > 0$, the conditional probability density function of $X$ given $Y = y$ is $f_{X\mid Y=y}(x) = y e^{-yx}, x > 0$. If the marginal probability density function of $Y$ is $g(y) = y e^{-y}, y > 0$ then $E(Y\mid X = 1) =$ ______________ (correct up to one decimal place).

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9

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let $(X, Y)$ be a random vector with the joint moment generating function $M_{X,Y}(s, t) = e^{s^2 + st^2 + t}, -\infty < s, t < \infty$. Let $\Phi(\cdot)$ denote the distribution function of the standard normal distribution and $p = P(X + 2Y < 1)$. If $\Phi(0) = 0.5, \Phi(0.5) = 0.6915, \Phi(1) = 0.8413$ and $\Phi(1.5) = 0.9332$ then the value of $2 p + 1$ (round off to two decimal places) equals ______________

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10

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let \((X, Y)\) be a random vector with joint probability mass function \[ f_{X,Y}(x, y) = \begin{cases} ^x C_y \left(\frac{1}{4}\right)^x, & y = 0,1,2,...,x; \; x = 1,2,..., \\ 0, & \text{otherwise} \end{cases} \] where \[ ^x C_y = \frac{x!}{y!(x-y)!}. \] Then the variance of \(Y\) equals ____________________

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11

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let the random vector \((X, Y)\) have the joint distribution function \[ F(x, y) = \begin{cases} 0, & x < 0 \text{ or } y < 0 \\ \frac{1 - e^{-x}}{4}, & x \ge 0, 0 \le y < 1 \\ 1 - e^{-x}, & x \ge 0, y \ge 1 \end{cases}. \] Let \(\text{Var}(X)\) and \(\text{Var}(Y)\) denote the variances of random variables \(X\) and \(Y\), respectively. Then \[ 16 \,\text{Var}(X) + 32 \,\text{Var}(Y) = ____________________ \]

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12

2020 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2020

Let the random vector \(\underline{X} = (X_1, X_2, X_3)\) have the joint probability density function \[ f_{\underline{X}}(x_1, x_2, x_3) = \begin{cases} \frac{81}{4} x_1^2 x_2^2 x_3^2, & -1 \le x_1 \le x_2 \le x_3 \le 1, \\ 0, & \text{otherwise} \end{cases}. \] Then the variance of the random variable \(X_1 + X_2 + X_3\) equals ____________________ (correct up to one decimal place).

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13

2021 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2021

Let \( (X, Y) \) have the joint probability density function \[ f_{X,Y}(x,y) = \begin{cases} \frac{4}{(x+y)^3}, & x > 1, y > 1, \\ 0, & \text{otherwise}. \end{cases} \] Then which one of the following statements is NOT true?

A

The probability density function of \(X + Y \) is \(f_{X+Y}(z) = \begin{cases} \frac{4}{z^3} (z-2), & z > 2, \\ 0, & \text{otherwise}. \end{cases}\)

B

\(P(X + Y > 4) = \frac{3}{4} \)

C

\(E(X + Y) = 4 \log_e 2 \)

D

\(E(Y \mid X = 2) = 4 \)

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14

2021 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2021

Let \( X_1, X_2 \) and \( X_3 \) be three uncorrelated random variables with common variance \( \sigma^2 < \infty \). Let \( Y_1 = 2X_1 + X_2 + X_3 \), \( Y_2 = X_1 + 2X_2 + X_3 \) and \( Y_3 = X_1 + X_2 + 2X_3 \). Then which of the following statements is/are true? P : The sum of eigenvalues of the variance covariance matrix of \( (Y_1, Y_2, Y_3) \) is \( 18\sigma^2 \). Q : The correlation coefficient between \( Y_1 \) and \( Y_2 \) equals that between \( Y_2 \) and \( Y_3 \).

A

P only

B

Q only

C

Both P and Q

D

Neither P nor Q

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15

2021 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2021

Let \((X, Y)\) have a bivariate normal distribution with the joint probability density function
\(f_{XY}(x, y) = \frac{1}{\pi} e^{\left(\frac{3}{2}xy - \frac{25}{32}x^2 - 2y^2\right)}, \ -\infty < x, y < \infty.\).
Then \(8 E(XY)\) equals __________

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16

2022 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2022

Let \( (X, Y, Z) \) be a random vector with the joint probability density function
\( f_{X,Y,Z}(x, y, z) = \begin{cases} \frac{1}{3}(2x + 3y + z), & 0 < x < 1, 0 < y < 1, 0 < z < 1, \\ 0, & \text{elsewhere}. \end{cases} \)
Then which one of the following points is on the regression surface of \( X \) on \( (Y, Z) \)?

A

\(\left( \frac{4}{7}, \frac{1}{3}, \frac{1}{3} \right) \)

B

\(\left( \frac{6}{7}, \frac{2}{3}, \frac{2}{3} \right) \)

C

\(\left( \frac{1}{2}, \frac{1}{3}, \frac{2}{3} \right) \)

D

\(\left( \frac{1}{2}, \frac{2}{3}, \frac{1}{3} \right) \)

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17

2022 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2022

Let \(X\) and \(Y\) be random variables such that \(X\) is uniformly distributed over \((0, 4)\), and the conditional distribution of \(Y\) given \(X = x\) is uniformly distributed over \(\left(0, \frac{x}{4}\right)\). Then \(E(Y^2)\) (rounded off to three decimal places) is equal to ________

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18

2023 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2023

Let \((X, Y)\) have joint probability density function
\(f(x, y) = \begin{cases} 8xy & \text{if } 0 < x < y < 1 \\ 0 & \text{otherwise}. \end{cases}\) If \(E(X \mid Y = y_0) = \frac{1}{2}\), then \(y_0\) equals

A

\(\frac{3}{4}\)

B

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

C

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

D

\(\frac{2}{3}\)

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19

2023 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2023

Suppose that there are 5 boxes, each containing 3 blue pens, 1 red pen and 2 black pens. One pen is drawn at random from each of these 5 boxes. If the random variable \(X_1\) denotes the total number of blue pens drawn and the random variable \(X_2\) denotes the total number of red pens drawn, then \(P(X_1 = 2, X_2 = 1)\) equals

A

\(\frac{5}{36}\)

B

\(\frac{5}{18}\)

C

\(\frac{5}{12}\)

D

\(\frac{5}{9}\)

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20

2023 · Statistics · Joint Distributions · Joint, Marginal and Conditional Distributions

Statistics (ST) 2023

Let \((X, Y)\) have joint probability mass function
\[p(x,y) = \begin{cases} \frac{c}{2^{x+y+2}} & \text{if } x = 0, 1, 2, \ldots; \ y = 0, 1, 2, \ldots; \ x \neq y \\ 0 & \text{otherwise.} \end{cases}\]
Then which one of the following statements is true?

A

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

B

\(c = \frac{1}{4}\)

C

c > 1

D

X and Y are independent

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