Multivariate Analysis-Multivariate Normal Distribution (ISS (Statistical Services) 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

Equation

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

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

Explanation

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

Equation

The mean is

Equation

Equation

The variance-covariance matrix is

Equation

Equation

Equation

Equation

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

(ii) X 2 and Equation are independent if

Equation

Equation

Equation

Equation

Equation

Equation

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

Equation

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

Equation

where these terms are defined by this equation

Equation

Given that

Equation

So, the value of equation is

Equation

Equation

Equation

The partial correlation is

Equation

Equation

Equation

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 Equation population with Equation , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for Equation for all Equation .

Explanation

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

Equation

From, the theorem of linear combinations of multivariate normal distribution is that every linear combination of X follows an univariate normal distribution.

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