Multivariate normal distribution - Multivariate Analysis - Statistics Previous Year Questions

1

2019 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2019

Let \((X_1, X_2, X_3)\) be a trivariate normal random vector with mean vector \((-3, 1, 4)\) and variance-covariance matrix \[ \begin{pmatrix} 4 & 0 & 0 \\ 0 & 3 & -3 \\ 0 & -3 & 4 \end{pmatrix} \]. Which of the following statements are true? (i) \(X_2\) and \(X_3\) are independent. (ii) \(X_1 + X_3\) and \(X_2\) are independent. (iii) \((X_2, X_3)\) and \(X_1\) are independent. (iv) \(\frac{1}{2}(X_2 + X_3)\) and \(X_1\) are independent.

A

(i) and (iii)

B

(ii) and (iii)

C

(i) and (iv)

D

(iii) and (iv)

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2

2019 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2019

Suppose that \(P_1\) and \(P_2\) are two populations with equal prior probabilities having bivariate normal distributions with mean vectors \((2,3)\) and \((1,1)\), respectively. The variance-covariance matrix of both the distributions is the identity matrix. Let \(z_1 = (2.5, 2)\) and \(z_2 = (2, 1.5)\) be two new observations. According to Fisher’s linear discriminant rule,

A

\(z_1\) is assigned to \(P_1\), and \(z_2\) is assigned to \(P_2\)

B

\(z_1\) is assigned to \(P_2\), and \(z_2\) is assigned to \(P_1\)

C

\(z_1\) is assigned to \(P_1\), and \(z_2\) is assigned to \(P_1\)

D

\(z_1\) is assigned to \(P_2\), and \(z_2\) is assigned to \(P_2\)

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3

2019 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2019

Let \((X_1, X_2)\) be a random vector with variance-covariance matrix \(\begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}\). The two principal components are

A

\(X_1\) and \(X_2\)

B

\(-X_1\) and \(X_2\)

C

\(X_1\) and \(-X_2\)

D

\(X_1 + X_2\) and \(X_2\)

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4

2019 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2019

The maximum likelihood estimates of the mean vector and the variance-covariance matrix of a bivariate normal distribution based on the realization \(\left(\binom{1}{2}, \binom{3}{4}, \binom{4}{3}\right)\) of a random sample of size 3, are given by

A

\(\binom{3}{3}\) and \(\begin{pmatrix} 2 & 1 \\ 1 & 2/3 \end{pmatrix}\)

B

\(\binom{3}{3}\) and \(\begin{pmatrix} 2 & 1 \\ 1 & 3/2 \end{pmatrix}\)

C

\(\binom{3}{3}\) and \(\begin{pmatrix} 3 & 3/2 \\ 3/2 & 2/3 \end{pmatrix}\)

D

\(\binom{3}{3}\) and \(\begin{pmatrix} 3 & 2/3 \\ 2/3 & 1 \end{pmatrix}\)

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5

2020 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2020

Let \(\underline{X}_1, \ldots, \underline{X}_n\) be a random sample of size \(n \ (\geq 2)\) from \(N_p(\underline{0}, \Sigma)\) distribution, where \(1 \leq p \leq n - 1\) and \(\Sigma\) is a positive definite matrix. Define \(\bar{\underline{X}} = \frac{1}{n} \sum_{i=1}^n \underline{X}_i \ \text{ and } \ (n - 1)S = \sum_{i=1}^n (\underline{X}_i - \bar{\underline{X}})(\underline{X}_i - \bar{\underline{X}})^T,\) where for any column vector \(\underline{U}, \underline{U}^T\) denotes its transpose. Then the distribution of the statistic \(T^2 = n \bar{\underline{X}}^T S^{-1} \bar{\underline{X}}\) is

A

\(\chi^2_p\), the central chi-square distribution with \(p\) degrees of freedom

B

\(F_{p, n-p}\), the central \(F\) distribution with \(p\) and \(n-p\) degrees of freedom

C

\(\frac{p(n-1)}{n-p} F_{p, n-p}\), where \(F_{p, n-p}\) is the central \(F\) distribution with \(p\) and \(n-p\) degrees of freedom

D

\(\frac{n-p}{p(n-1)} F_{n-p, p}\), where \(F_{n-p, p}\) is the central \(F\) distribution with \(n-p\) and \(p\) degrees of freedom

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6

2020 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2020

Let the random vector \( \underline{X} = (X_1, X_2, X_3, X_4) \) follow \( N_4(\underline{\mu}, \Sigma) \) distribution, where
\[ \underline{\mu} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \text{ and } \Sigma = \begin{bmatrix} 1 & 0.7 & 0.6 & 0.4 \\ 0.7 & 1 & 0.5 & 0.8 \\ 0.6 & 0.5 & 1 & 0.7 \\ 0.4 & 0.8 & 0.7 & 1 \end{bmatrix} \]
Then
\( P(X_1 + X_2 + X_3 + X_4 > 0) = \) ___________ (correct up to one decimal place).

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7

2020 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2020

Let \( \underline{X} \) be a \( 4 \times 1 \) random vector with \( E(\underline{X}) = \underline{0} \) and variance-covariance matrix
\[ \Sigma = \begin{bmatrix} 1 & 0.5 & 0.5 & 0.5 \\ 0.5 & 1 & 0.5 & 0.5 \\ 0.5 & 0.5 & 1 & 0.5 \\ 0.5 & 0.5 & 0.5 & 1 \end{bmatrix} \]
Let \( \underline{Y} \) be the \( 4 \times 1 \) random vector of principal components derived from \( \Sigma \). The proportion of total variation explained by the first two principal components equals ___________ (correct up to two decimal places).

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8

2020 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2020

Suppose that \( P_1 \) and \( P_2 \) are two populations having bivariate normal distributions with mean vectors \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \) and \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \), respectively, and the same variance-covariance matrix \( \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix} \). Let \( Z_1 = \begin{pmatrix} 0.75 \\ 0.75 \end{pmatrix} \) and \( Z_2 = \begin{pmatrix} 0.25 \\ 0.25 \end{pmatrix} \) be two new observations. If the prior probabilities for \( P_1 \) and \( P_2 \) are assumed to be equal and the misclassification costs are also assumed to be equal then, according to linear discriminant rule,

A

\(Z_1 \) is assigned to \(P_1 \) and \(Z_2 \) is assigned to \(P_2 \)

B

\(Z_1 \) is assigned to \(P_2 \) and \(Z_2 \) is assigned to \(P_1 \)

C

both \(Z_1 \) and \(Z_2 \) are assigned to \(P_1 \)

D

both \(Z_1 \) and \(Z_2 \) are assigned to \(P_2 \)

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9

2020 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2020

Let \( \underline{X}_1, \underline{X}_2, ..., \underline{X}_{10} \) be independent and identically distributed \( N_3(\underline{0}, I_3) \) random vectors, where \( I_3 \) is the \( 3 \times 3 \) identity matrix. Let \( T = \sum_{i=1}^{10} \left( \underline{X}_i^T \left( I_3 - \frac{1}{3} J_3 \right) \underline{X}_i \right) \), where \( J_3 \) is the \( 3 \times 3 \) matrix with each entry 1 and for any column vector \( \underline{U} \), \( \underline{U}^T \) denotes its transpose. Then the distribution of \( T \) is

A

central chi-square with 5 degrees of freedom

B

central chi-square with 10 degrees of freedom

C

central chi-square with 20 degrees of freedom

D

central chi-square with 30 degrees of freedom

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10

2020 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2020

Let the joint distribution of the random variables $X_1, X_2$ and $X_3$ be $N_3(\mu, \Sigma)$, where $\mu = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ and $\Sigma = \begin{bmatrix} 1 & 0.5 & 0 \\ 0.5 & 1 & 0 \\ 0 & 0 & 5 \end{bmatrix}$. Then which of the following statements is TRUE?

A

$X_1 - X_2 + X_3$ and $X_1$ are independent

B

$X_1 + X_2$ and $X_3 - X_1$ are independent

C

$X_1 - X_2 + X_3$ and $X_1 + X_2$ are independent

D

$X_1 - 2X_2$ and $2X_1 + X_2$ are independent

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11

2021 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2021

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 \( -1 < \rho < 1 \). Let \( \Phi_{\rho}(0, 0) = P(X \leq 0, Y \leq 0) \). Then the Kendall's \( \tau \) coefficient between \( X \) and \( Y \) equals

A

\(4\Phi_{\rho}(0, 0) - 1 \)

B

\(4\Phi_{\rho}(0, 0) \)

C

\(4\Phi_{\rho}(0, 0) + 1 \)

D

\(\Phi_{\rho}(0, 0) \)

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12

2021 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2021

Let \(X_1, X_2\) and \(X_3\) be a random sample from a bivariate normal distribution with unknown mean vector \(\mu\) and unknown variance-covariance matrix \(\Sigma\), which is a positive definite matrix. The \(p\)-value corresponding to the likelihood ratio test for testing \(H_0: \mu = \underline{0}\) against \(H_1: \mu \neq \underline{0}\) based on the realization \(\left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 4 \\ -2 \end{pmatrix}, \begin{pmatrix} -5 \\ 0 \end{pmatrix}\right\}\) of the random sample equals __________ (round off to 2 decimal places).

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13

2021 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2021

Let the joint distribution of random variables \( X_1, X_2, X_3 \) and \( X_4 \) be \( N_4(\underline{\mu}, \Sigma) \), where \[ \underline{\mu} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \text{ and } \Sigma = \begin{bmatrix} 1 & 0.2 & 0 & 0 \\ 0.2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0.2 \\ 0 & 0 & 0.2 & 1 \end{bmatrix}. \] Then which one of the following statements is true?

A

\(\frac{5}{17}[(X_1 + X_2)^2 + (X_3 + X_4 - 1)^2] \) follows a central chi-square distribution with \(2 \) degrees of freedom

B

\(\frac{1}{3}[(X_1 + X_3 - 1)^2 + (X_2 + X_4 - 1)^2] \) follows a central chi-square distribution with \(2 \) degrees of freedom

C

\(E\left[ \sqrt{\frac{|X_1 + X_2 - 1|}{X_3 + X_4 - 1}} \right] \) is NOT finite

D

\(E\left[ \left| \frac{X_1 + X_2 + X_3 + X_4 - 2}{X_1 + X_2 - X_3 - X_4} \right| \right] \) is NOT finite

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14

2022 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2022

Let \((3,6)^T, (4,4)^T, (5,7)^T\) and \((4,7)^T\) be four independent observations from a bivariate normal distribution with the mean vector \(\boldsymbol{\mu}\) and the covariance matrix \(\boldsymbol{\Sigma}\). Let \(\hat{\boldsymbol{\mu}}\) and \(\hat{\boldsymbol{\Sigma}}\) be the maximum likelihood estimates of \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}\), respectively, based on these observations. Then \(\hat{\boldsymbol{\Sigma}} \hat{\boldsymbol{\mu}}\) is equal to

A

\(\begin{pmatrix} 3.5 \\ 10 \end{pmatrix}\)

B

\(\begin{pmatrix} 7.5 \\ 4 \end{pmatrix}\)

C

\(\begin{pmatrix} 4 \\ 13.5 \end{pmatrix}\)

D

\(\begin{pmatrix} 10 \\ 3.5 \end{pmatrix}\)

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15

2022 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2022

Let \(\mathbf{X} = (X_1, X_2, X_3)^T\) be a random vector with the distribution \(N_3(\boldsymbol{\mu}, \boldsymbol{\Sigma})\), where \[ \boldsymbol{\mu} = \begin{pmatrix} 3 \\ 2 \\ 4 \end{pmatrix} \quad \text{and} \quad \boldsymbol{\Sigma} = \begin{pmatrix} 4 & 2 & 1 \\ 2 & 3 & 0 \\ 1 & 0 & 2 \end{pmatrix} \] Then \(E(X_1 | X_2 = 4, X_3 = 7)\) (in integer) is equal to ________

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16

2023 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2023

Suppose that \(\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots, \boldsymbol{X}_n, \boldsymbol{Y}_1, \boldsymbol{Y}_2, \ldots, \boldsymbol{Y}_n\) are independent and identically distributed random vectors each having \(N_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) distribution, where \(\boldsymbol{\Sigma}\) is non-singular, \(p > 1\) and \(n > 1\). If \(\bar{\boldsymbol{X}} = \frac{1}{n} \sum_{i=1}^n \boldsymbol{X}_i\) and \(\bar{\boldsymbol{Y}} = \frac{1}{n} \sum_{i=1}^n \boldsymbol{Y}_i\), then which one of the following statements is true?

A

There exists \(c > 0\) such that \(c (\bar{\boldsymbol{X}} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\bar{\boldsymbol{X}} - \boldsymbol{\mu})\) has \(\chi^2\)-distribution with \(p\) degrees of freedom

B

There exists \(c > 0\) such that \(c (\bar{\boldsymbol{X}} - \bar{\boldsymbol{Y}})^T \boldsymbol{\Sigma}^{-1} (\bar{\boldsymbol{X}} - \bar{\boldsymbol{Y}})\) has \(\chi^2\)-distribution with \((p - 1)\) degrees of freedom

C

There exists \(c > 0\) such that \(c \sum_{i=1}^n (\boldsymbol{X}_i - \bar{\boldsymbol{X}})^T \boldsymbol{\Sigma}^{-1} (\boldsymbol{X}_i - \bar{\boldsymbol{X}})\) has \(\chi^2\)-distribution with \(p\) degrees of freedom

D

There exists \(c > 0\) such that \(c \sum_{i=1}^n (\boldsymbol{X}_i - \boldsymbol{Y}_i - \bar{\boldsymbol{X}} + \bar{\boldsymbol{Y}})^T \boldsymbol{\Sigma}^{-1} (\boldsymbol{X}_i - \boldsymbol{Y}_i - \bar{\boldsymbol{X}} + \bar{\boldsymbol{Y}})\) has \(\chi^2\)-distribution with \(p\) degrees of freedom

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17

2023 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2023

Suppose that \( (X_1, X_2, X_3) \) has \( N_3(\boldsymbol{\mu}, \Sigma) \) distribution with \( \boldsymbol{\mu} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \) and
\( \Sigma = \begin{bmatrix} 2 & 2 & 1 \\ 2 & 5 & 1 \\ 1 & 1 & 1 \end{bmatrix} \).
Given that \( \Phi(-0.5) = 0.3085 \), where \( \Phi(\cdot) \) denotes the cumulative distribution function of a standard normal random variable,
\( P \left( (X_1 - 2X_2 + 2X_3)^2 < \frac{7}{2} \right) \) (rounded off to two decimal places) equals ______________

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18

2023 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2023

Suppose that \( X_1, X_2, \ldots, X_{10} \) are independent and identically distributed random vectors each having \( N_2(\boldsymbol{\mu}, \Sigma) \) distribution, where \( \Sigma \) is non-singular. If \( U = \frac{1}{1 + (\bar{X} - \boldsymbol{\mu})^T \Sigma^{-1} (\bar{X} - \boldsymbol{\mu})} \), where \( \bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i \), then the value of \( \log_e P \left( U \leq \frac{1}{2} \right) \) equals

A

\(-5 \)

B

\(-10 \)

C

\(-2 \)

D

\(-1 \)

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19

2024 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2024

Let \((X_1, X_2, X_3)\) have \(N_3(\boldsymbol{\mu}, \Sigma)\) distribution with \(\boldsymbol{\mu} = \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}\) and \(\Sigma = \begin{bmatrix} 25 & -2 & 4 \\ -2 & 4 & 1 \\ 4 & 1 & 9 \end{bmatrix}\). For which of the following vectors \(\boldsymbol{a}\), \(X_2\) and \(X_2 - \boldsymbol{a}^T \begin{bmatrix} X_1 \\ X_2 \end{bmatrix}\) are independent?

A

\(\boldsymbol{a} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

B

\(\boldsymbol{a} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}\)

C

\(\boldsymbol{a} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}\)

D

\(\boldsymbol{a} = \begin{bmatrix} 2 \\ 2 \end{bmatrix}\)

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20

2025 · Statistics · Multivariate Analysis · Multivariate normal distribution

Statistics (ST) 2025

Let \(X_1, X_2, ..., X_5\) be i.i.d. random vectors following the bivariate normal distribution with zero mean vector and identity covariance matrix. Define \(5 \times 2\) matrix \(X\) as \(X = (X_1, X_2, ..., X_5)^T\). Further, let \(W = (W_{ij}) = X^T X\), and \(Z = W_{11} + 4W_{12} + 4W_{22}\). Then Var(\(Z\)) equals

A

150

B

200

C

250

D

300

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