Question

(4) Suppose \( Y=\left(Y_{1}, Y_{2}, Y_{3}, Y_{4}\right)^{T} \) follows \( N_{4}(\mu, \Sigma) \), where \[ \mu=\left(\begin{array}{c} 1 \\ 2 \\ 3 \\ -2 \end{array}\right), \Sigma=\left(\begin{array}{cccc} 4 & 2 & -1 & 2 \\ 2 & 6 & 3 & -2 \\ -1 & 3 & 5 & -4 \\ 2 & -2 & -4 & 4 \end{array}\right) \text {. } \] Find the following : (a) The marginal distribution of \( \left(Y_{1}, Y_{3}\right)^{T} \). (b) The marginal distribution of \( Y_{2} \). (c) The distribution of \( Z=Y_{1}+2 Y_{2}-Y_{3}+3 Y_{4} \). (d) The conditional distribution of \( X_{1}=\left(Y_{1}, Y_{2}\right)^{T} \) given that \( X_{2}=\left(Y_{3}, Y_{4}\right)^{T}= \) \( \left(y_{3}, y_{4}\right)^{T}=x_{2} \).

          (4) Suppose \( Y=\left(Y_{1}, Y_{2}, Y_{3}, Y_{4}\right)^{T} \) follows \( N_{4}(\mu, \Sigma) \), where
\[
\mu=\left(\begin{array}{c}
1 \\
2 \\
3 \\
-2
\end{array}\right), \Sigma=\left(\begin{array}{cccc}
4 & 2 & -1 & 2 \\
2 & 6 & 3 & -2 \\
-1 & 3 & 5 & -4 \\
2 & -2 & -4 & 4
\end{array}\right) \text {. }
\]

Find the following :
(a) The marginal distribution of \( \left(Y_{1}, Y_{3}\right)^{T} \).
(b) The marginal distribution of \( Y_{2} \).
(c) The distribution of \( Z=Y_{1}+2 Y_{2}-Y_{3}+3 Y_{4} \).
(d) The conditional distribution of \( X_{1}=\left(Y_{1}, Y_{2}\right)^{T} \) given that \( X_{2}=\left(Y_{3}, Y_{4}\right)^{T}= \) \( \left(y_{3}, y_{4}\right)^{T}=x_{2} \).

(4) Suppose Y=(Y1, Y2, Y3, Y4)^T follows N4(μ, Σ), where

μ=(
1

2

3

-2
), Σ=(
4 2 -1 2

2 6 3 -2

-1 3 5 -4

2 -2 -4 4
) .

Find the following :
(a) The marginal distribution of (Y1, Y3)^T.
(b) The marginal distribution of Y2.
(c) The distribution of Z=Y1+2 Y2-Y3+3 Y4.
(d) The conditional distribution of X1=(Y1, Y2)^T given that X2=(Y3, Y4)^T= (y3, y4)^T=x2.

Added by Jessica W.

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