# Multivariate Analysis-Multivariate Normal Distribution (ISS Statistics Paper II (Old Subjective Pattern)): Questions 1 - 3 of 3

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## Question number: 1

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let X= (X 1, X 2, X 3) ’ be distributed as N 3 (µ, ∑) where µ’= (2, -3, 1) and

(i) Find the distribution of 3X 1 -2X 2 +X 3.

(ii) Find a 2 × 1 vector a such that X 2 and are independent.

### Explanation

(i) the distribution of 3X 1 -2X 2 +X 3 is

The mean is

The variance-covariance matrix is

So, the Y has a normal distribution N (13, 9).

(ii) X 2 and are independent if

## Question number: 2

» Multivariate Analysis » Multivariate Normal Distribution » Partial Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let X = (X 1, X 2, X 3) ’ be distributed as N 3 (µ, ∑) where µ = (10, -7, 2) ’ and

Find the partial correlation between X 1 and X 2 given X 3.

### Explanation

The partial correlation between X 1 and X 2 given X 3 is defined as

where these terms are defined by this equation

Given that

So, the value of equation is

The partial correlation is

## Question number: 3

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let X 1, X 2, …, X n be a random sample from an population with , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for for all .

### Explanation

Let X 1, X 2, …, X n be a random sample from an and assume the linear combination of the random sample is

From, the theorem of linear combinations of multivariate normal distribution is that every linear combination of X follows an univariate normal distribution.… (91 more words) …

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