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Question:

Let $X_1, X_2, \dots, X_n$ be a random sample from a population with mean $\mu$ and variance $\sigma^2$ . Consider the estimator
$$\hat{\mu} = \frac{1}{n} \sum_{i=1}^{n} X_i + 2$ $
(i) Find the bias of the estimator $\hat{\mu}$ .
(ii) Find the variance of $\hat{\mu}$ .
(iii) Hence, calculate the Mean Square Error (MSE) of $\hat{\mu}$ .

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Question:

A school caretaker looks after 6 children. The ages (in years) of the children are given below:

Child	Age (in years)
Meena	5
Aarav	4
Nisha	6
Kabir	4
Pooja	5

(i) What is the nature (type) of the population of the ages of children?
(ii) Construct the sampling distribution of the sample mean for all possible samples of size n = 2.
(iii) Comment on whether the sampling distribution is approximately normal.
(iv) Find the mean and standard error of the sampling distribution. (20)

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Question:

State whether the following statements are True or False. Give reasons in support of your answer:

(i) The mean square error is calculated when the estimator is unbiased.

(ii) If the sample size tends to infinity, the variance of a consistent estimator must tend to zero.

(iii) In the method of moments, population moments are equated to corresponding sample moments.

(iv) A 95% confidence interval is smaller than a 99% confidence interval.

(v) The chi-square distribution is symmetric for all degrees of freedom.

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Question:

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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Question:

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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Question:

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

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Question:

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).

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Question:

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

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

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Question:

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 $\text{X}^{(2)} | \text{X}^{(1)} = \text{x}^{(1)}$ .

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Question:

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

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Question:

एक खुदरा (रिटेल) कंपनी के विभिन्न विभागों में डेटा के इस्तेमाल का उल्लेख कीजिए।

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Question:

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Question:

खुदरा (रिटेल) अंतः क्रियाओं में संचार के विनिमय सिद्धांत (Exchange Theory of Communication) को कैसे लागू किया जाता है?

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Question:

प्रौद्योगिकी-सक्षम संचार (Technology Enabled Communication) के सकारात्मक और नकारात्मक प्रभावों पर चर्चा कीजिए।

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Question:

ग्राहक संचार प्रबंधन (Customer Communication Management) खुदरा विक्रेता की ब्रांडिंग को कैसे बेहतर बनाने में मदद कर सकता है?

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Question:

स्वेच्छाचारी (Autocratic), लोकतांत्रिक (Democratic) और अहस्तक्षेपकारी / लैसेज़-फेयर (Laissez Faire) नेतृत्व शैलियों में अंतर स्पष्ट कीजिए।

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Question:

सीमांत विश्लेषण (Marginal Analysis) और लागत-लाभ विश्लेषण (Cost-Benefit Analysis) विकल्पों के मूल्यांकन के उपकरण के रूप में कैसे कार्य करते हैं?

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Question:

विभिन्न प्रकार की योजनाएँ (Plans) कौन-कौन सी हैं?

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Question:

लाइन और स्टाफ कार्यों में क्या अंतर है? कंपनी जिन विभिन्न प्रकार की संगठनात्मक संरचनाओं पर विचार कर सकती है, उनके संदर्भ में इसकी चर्चा कीजिए।

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