Solve your IGNOU Doubts

Question:

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

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Question:

Relationship between (ii) and (iii).

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Question:

Clustering

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Question:

Hotelling’s T2

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Question:

Mahalanobis D2

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Question:

Covariance Matrix

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Question:

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

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Question:

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 $\underline{x}=\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}\quad\text{and}\quad\underline{y}=\begin{pmatrix}y_1\\y_2\\y_3\\y_4\end{pmatrix}$ then obtain the sample covariance matrix between $\underline{x}\text{and}\underline{y}$

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

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Question:

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

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

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Question:

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

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

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

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Question:

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

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

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Question:

$\text{Let}\underline{X}=\begin{pmatrix}X_1\\X_2\end{pmatrix}$ has the following joint density function

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

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

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Question:

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 $\underline{X}\text{and}\underline{Y}$ is symmetric.

(b) $if\underline{x}$ 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) $\text{If}\underline{X}\sim N_2\left(\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right)\text{and}\underline{Y}\sim N_2\left(\begin{pmatrix}-1\\3\end{pmatrix},\begin{pmatrix}1&0\\0&1\end{pmatrix}\right),\text{then}$

$\underline{X}+\underline{Y}\sim N_2\left(\begin{pmatrix}1\\4\end{pmatrix},\begin{pmatrix}2&0\\0&2\end{pmatrix}\right).$

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Question:

Define the Residual B.I.B.D. and Derived B.I.B.D. of a given symmetric B.I.B.D. Mention the rule of constructing a residual B.I.B.D. corresponding to a specific B.I.B.D.

$\text{Let}\mathbf{X}=\{1,2,3,4,5\}\text{and}\mathcal{A}=\{\{1,2,3\},\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,3,5\},\{1,4,5\},\dots\}$

$\{\{2,3,4\},\{2,3,5\},\{2,4,5\},\{3,4,5\}\}$ then (X, ) is a (5, 3, 3)- B.I.B.D. Find the incidence matrix of this B.I.B.D.

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Question:

Let us consider the following Balanced Incomplete Block Design (B.I.B.D.) with parameters $v=9,\quad b=7,\quad r=7,\quad k=4,\quad\lambda=2$

Block Label	Design
I	1 3 4 5
II	1 4 6 7
III	1 2 5 7
IV	3 5 6 7
V	2 3 4 7
VI	1 2 3 6
VII	2 4 5 6

Obtain a derived design from the above Balanced Incomplete Block Design (B.I.B.D.) and find the parameters of the obtained design.

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Question:

Explain what is meant by a one-quarter fractional factorial experiment of a 2 ² factorial experiment. Give a notation which denotes the one-quarter fractional factorial design.

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Question:

Below is given the plan and yields of 2²-Factorial Experiment involving 2 factors A and B each at two levels 0 and 1. Analyse the design

Block I
117 (1)	106 b	129 ab	124 a
Block II
124 ab	120 (1)	117 b	124 a
Block III
111 (1)	127 a	114 b	126 ab
Block IV
125 ab	131 a	112 b	108 (1)
Block V
95 ab	97 b	73 (1)	128 a
Block VI
158 a	81 (1)	125 ab	117 b

Does treatment effect A differs from treatment effect B?

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Question:

Suggest an estimator of population mean in cluster sampling with unequal size clusters, which is based upon the means of selected clusters. Determine whether it is an unbiased estimator?

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Question:

A population consisting of 6 clusters, each of size 6 is given below. The values of the study variable Y, noted on each of the units within each cluster are also provided. A random sample of size 3 clusters was selected from the population and 3 elementary units from the selected clusters were randomly chosen

Cluster	Y - values	Cluster	Y - values
1	2, 4, 6, 1, 3, 5	4	3, 2, 5, 1, 6, 4
2	2, 5, 3, 4, 7, 4	5	2, 4, 6, 8, 3, 5
3	4, 3, 6, 2, 1, 5	6	4, 1, 2, 7, 5, 3

Let the 2 nd, 4th and 6thclusters be selected randomly in the first-stage sample. Further, let the y-values 2, 7, 3 of the 2nd cluster; 6, 3, 1 of the 4th cluster and 7, 4, 3 of the 6th cluster be selected randomly for the second-stage sample

Estimate the population mean on the basis of the suggested estimator and compare it with the actual value of the population mean

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Question:

Define the difference estimator for the population mean. Show that it is an unbiased estimator and its variance is given by $V(\bar{Y}_D)=\frac{N-n}{Nn}S_Y^2(1-\rho^2)$ where symbols have their usual meanings.