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MST 18: Multivariate Analysis

Title Name	IGNOU MST 18 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCAST
Course Name	M.Sc. (Applied Statistics)
Subject Code	MST 18
Subject Name	Multivariate Analysis
Year	2025
Session	-
Language	English Medium
Assignment Code	MST 18/Assignment-1/2025
Product Description	Assignment of MSCAST (M.Sc. (Applied Statistics)) 2025. Latest MST 018 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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MST 018 (January 2025 - July 2025) - ENGLISH

TUTOR MARKED ASSIGNMENT

MST-018: Multivariate Analysis

Course Code: MST-018

Assignment Code: MST-018/TMA/2025

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are true or false and also give the reason in support of your answer:

(a) The covariance matrix of random vectors $underline{X} ext{and}underline{Y}$ is symmetric.

(b) $ifunderline{x}$ is a p-variate normal random vector, then every linear combination $underline{c{}'}$ $underline{x}$ is a scalar vector, is also p-variate normal vector

(d) If a matrix is positive definite then its inverse is also positive definite.

(e) $ext{If}underline{X}sim N_2left(egin{pmatrix}2\1end{pmatrix},egin{pmatrix}1&0\0&1end{pmatrix} ight) ext{and}underline{Y}sim N_2left(egin{pmatrix}-1\3end{pmatrix},egin{pmatrix}1&0\0&1end{pmatrix} ight), ext{then}$

$underline{X}+underline{Y}sim N_2left(egin{pmatrix}1\4end{pmatrix},egin{pmatrix}2&0\0&2end{pmatrix} ight).$

2(a) $ext{Let}underline{X}=egin{pmatrix}X_1\X_2end{pmatrix}$ has the following joint density function

$f(x_1,x_2)=egin{cases}4x_1x_2&,0<x_1<1,0<x_2<1\0&, ext{otherwise}.end{cases}$

Find the marginal distributions, mean vector and variance-covariance matrix. Also, comment on the independence of X₁and X₂

(b) $underline{X}=egin{pmatrix}underline{X}^{(1)}\underline{X}^{(2)}end{pmatrix}sim N_4(underline{mu},underline{Sigma})$ whare $underline{mu}=egin{pmatrix}-4\1\4\0end{pmatrix},quadunderline{Sigma}=egin{pmatrix}2&2&|&3&0\2&1&|&2&0\--&--&--&--&--\3&2&|&2&1\0&0&|&1&1end{pmatrix}$ Find the

$E(underline{X}^{(2)}|underline{X}^{(1)}=underline{x}^{(1)})quad ext{and}quad Cov(underline{X}^{(2)}|underline{X}^{(1)}=underline{x}^{(1)})$

3 (a) Let X  be a 3-dimensional random vector with dispersion matrix

$underline{Sigma}=egin{pmatrix}4&-2&0\-2&4&0\0&0&2end{pmatrix}$

Determine the first principal component and the proportion of the total variability that it explains.

(b) $underline{X}sim N_4(underline{mu},underline{Sigma}),quad ext{where}quadunderline{mu}=egin{pmatrix}3\-2\1\-2end{pmatrix}$ and $underline{Sigma}=egin{pmatrix}4&0&0&0\0&3&0&0\0&0&2&-2\0&0&-2&5end{pmatrix}$ Check the

independence $ext{of the(i)}X_2 ext{and}X_1quad ext{(ii)}(X_2,X_4) ext{and}(X_1,X_3)quad ext{(iii)}(X_1,X_2) ext{and}(X_3,X_4)$

4 (a) Consider the following data of 11 samples on 8 variables by Anscombe, Francis J. (1973):

x₁	x₂	x₃	x₄	y₁	y₂	y₃	y₄
10	10	10	8	8.04	9.14	7.46	6.58
8	8	8	8	6.95	8.14	6.77	5.76
13	13	13	8	7.58	8.74	12.74	7.71
9	9	9	8	8.81	8.77	7.11	8.84
11	11	11	8	8.33	9.26	7.81	8.47
14	14	14	8	9.96	8.10	8.84	7.04
6	6	6	8	7.24	6.13	6.08	5.25
4	4	4	19	4.26	3.10	5.39	12.50
12	12	12	8	10.84	9.13	8.15	5.56
7	7	7	8	4.82	7.26	6.42	5.56
5	7	5	8	5.68	4.74	5.73	5.56

If the vector $underline{x}=egin{pmatrix}x_1\x_2\x_3\x_4end{pmatrix}quad ext{and}quadunderline{y}=egin{pmatrix}y_1\y_2\y_3\y_4end{pmatrix}$ then obtain the sample covariance matrix between $underline{x} ext{and}underline{y}$

Source: Anscombe, Francis J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17– 21. doi: 10.2307/2682899.

(b) Obtain the maximum likelihood estimator of the mean vector and variance-covariance matrix of the multivariate normal distribution.

5 (a) Define the following:

(i) Covariance Matrix

(ii) Mahalanobis D2

(iii) Hotelling’s T2

(iv) Clustering

(v) Relationship between (ii) and (iii).

(b) $ext{If}underline{X}sim N_3(underline{mu},underline{Sigma}) ext{with}underline{mu}=egin{pmatrix}2\1\2end{pmatrix} ext{and}underline{Sigma}=egin{pmatrix}5&3&0\3&3&-2\0&-2&5end{pmatrix}.$ Then find the joint distribution of $X_1+2X_2,quad 2X_1-X_2quad ext{and}quad X_3.$

MST 018 (January 2026 - July 2026) - ENGLISH

**MST-018: Multivariate Analysis**

**Course Code: MST-018**
**Assignment Code: MST-018/TMA/2026**
**Maximum Marks: 100**

**Note: All questions are compulsory. Answer in your own words.**

1. State whether the following statements are True or False. Give reasons in support of your answer: **5×2=10**

(a) The eigenvalues of a positive definite matrix are less than zero.

(b) $A = \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & -1/\sqrt{2} \end{pmatrix}$ is an orthogonal and idempotent matrix.

(c) A factor model postulates that a random vector $\underline{X}$ is linearly dependent upon a few observable common factors.

(d) A given px1 vector $\underline{\mu}_0$ is a plausible value for the mean of a multivariate normal distribution can be tested using Student's t test.

(e) In single linkage method, we find the minimum distance in the distance matrix and merge the minimum distanced clusters.

**2 (a)** Let $\underset{\sim}{\text{X}}_{p \times 1}$ , $\underset{\sim}{\mu}_{p \times 1}$ and $\Sigma_{p \times p}$ be partitioned as $\underset{\sim}{\text{X}}_{p \times 1} = \begin{pmatrix} \text{X}_{k \times 1}^{(1)} \\ \text{X}_{(p-k) \times 1}^{(2)} \end{pmatrix}$ , $\underset{\sim}{\mu}_{p \times 1} = \begin{pmatrix} \mu_{k \times 1}^{(1)} \\ \mu_{(p-k) \times 1}^{(2)} \end{pmatrix}$ and

$\Sigma_{p \times p} = \begin{pmatrix} \Sigma_{11 \ k \times k} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$ . Then derive the conditional distribution of $\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)}$ .

---

**(b)** Let $\text{X} = \begin{pmatrix} \text{X}^{(1)} \\ \text{X}^{(2)} \end{pmatrix} \sim \text{N}_{4} (\mu, \Sigma)$ , where $\underset{\sim}{\mu} = \begin{pmatrix} -4 \\ 1 \\ \hline 4 \\ 0 \end{pmatrix}$ and $\Sigma = \left( \begin{array}{cc|cc} 2 & 0 & 1 & 0 \\ 0 & 2 & 2 & 0 \\ \hline 1 & 2 & 6 & 1 \\ 0 & 0 & 1 & 1 \end{array} \right)$ .

Then find $\text{E}(\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)})$ and $\text{Cov}(\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)})$ .

**15+10**

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**3 (a)** Let two independent samples of size $\text{N}_{1} = 10$ and $\text{N}_{2} = 12$ from trivariate normal populations have following mean vectors and variance-covariance matrices:

$\bar{\text{x}}_{1} = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}$ and $\text{S}_{1} = \begin{pmatrix} 1.2 & 0.3 & 0.2 \\ 0.3 & 1.5 & 0.4 \\ 0.2 & 0.4 & 1.8 \end{pmatrix}$

$\bar{\text{x}}_{2} = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}$ and $\text{S}_{2} = \begin{pmatrix} 1.0 & 0.2 & 0.3 \\ 0.2 & 1.4 & 0.3 \\ 0.3 & 0.3 & 1.6 \end{pmatrix}$ , respectively.

---
$$If \mathbf{S}_p^{-1} = \begin{pmatrix} 1.138 & -0.152 & -0.181 \\ -0.152 & 0.837 & -0.125 \\ -0.181 & -0.125 & 0.740 \end{pmatrix} \text{is the inverse of the pooled covariance matrix}$ $

$$\mathbf{S}_p = \begin{pmatrix} 1.09 & 0.245 & 0.255 \\ 0.245 & 1.445 & 0.345 \\ 0.255 & 0.345 & 1.69 \end{pmatrix}, \text{then test whether the mean vectors of the two}$ $

samples are significantly different from each other at 5% level of significance or not.
$$\text{(Use } F_{3, 18}(0.05) = 3.16 \text{ )}$ $

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**(b)** Let $\mathbf{X} \sim N_4 (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ , where $\boldsymbol{\mu} = \begin{pmatrix} 4 \\ -1 \\ 6 \\ -1 \end{pmatrix}$ and $\boldsymbol{\Sigma} = \begin{pmatrix} 5 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & -1 & 1 \end{pmatrix}$ . Check the independence

of (i) X₂ and X₁ (ii) (X₂, X₄) and (X₁, X₃) (iii) (X₁, X₂) and (X₃, X₄).
**15+10**

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**4 (a)** Obtain the square root matrix corresponding to a matrix $\mathbf{A} = \begin{pmatrix} 3 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & 1 & 3 \end{pmatrix}$ . Also

verify that $\mathbf{A}^{1/2} \mathbf{A}^{1/2} = \mathbf{A}$ .

**(b)** If $\mathbf{X} \sim N_3 (\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$ and $\boldsymbol{\Sigma} = \begin{pmatrix} 5 & 3 & 0 \\ 3 & 3 & -2 \\ 0 & -2 & 5 \end{pmatrix}$ . Then find the joint distribution

of X₁ + 2X₂, 2X₁ - X₂ and X₃.

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**5 (a) Define clustering. Differentiate between single linkage and complete linkage method of clustering.**

**(b) Consider the following partitioned sample variance-covariance matrix S, obtained from a sample of size 10.**

$$S = \left( \begin{array}{cc|cc} 4.8 & 2.9 & 0.2 & -0.2 \\ 2.9 & 7.9 & 0.0 & 0.7 \\ \hline 0.2 & 0.0 & 7.1 & 7.2 \\ -0.2 & 0.7 & 7.2 & 11.3 \end{array} \right)$ $

**If $|S| = 799.13$ , then test for the independence of the sub-vectors $X^{(1)}_{2 \times 1}$ and $X^{(2)}_{2 \times 1}$ .**
** $\quad$ (Use $\Lambda_{2, 2, 7}(0.05) = 0.230$ )**

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Would you like me to help you solve the hypothesis test for independence in part (b)?

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