Question

Let X \sim N_4 (\mu ,\Sigma ) with

\mu = \begin{pmatrix} 2 \\ 1 \\ 3 \\ -4 \end{pmatrix} and 

\Sigma \begin{bmatrix} 1 & 1 & 1 & 1\\ 1& 2& -2 & -1\\ 1& -2 & 9 & -1 \\ 1& -1 & -1 & 16 \end{bmatrix}

Support Y and Z are two partitioned subvectors of X such that Y' = (x1x3) and Z' = (x2x4)

i) Obtain the marginal distribution of Y′.

ii) Check the independence of  Y′ and Z′.

iii) Obtain the conditional distribution of Y′ | Z′ ; where Y' = (x1x2) , Z' = (x3x4)

iv) Find E(Y' | Z' )];  where  Y′ and  Z′ are same as in (iii).

06 Feb 2024
Answer :
Word Count : 694

i) To obtain the marginal distribution of \( Y' \), we need to extract the corresponding elements from \( \mu \) and \( \Sigma \). From \( \mu \), the mean vector of \( X \), we take the elements corresponding to \( x_1 \) and \( x_3 \), which are \( \mu_1 = 2 \) and \( \mu_3 = 3 \).

The covariance matrix \( \Sigma \) indicates the relationships between all elements of \( X \). To extract the covariance terms relevant to \( x_1 \) and \( x_3 \), we need the rows and columns corresponding to these variables. The covariance matrix for \( Y' \) is then:
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