Question

X=\binom{X^{(1)}}{X_{(2)}}\sim N_4\left ( \mu ,\sum \right ),  where \begin{pmatrix} -4\\ \frac{1}{4} \\ 0 \end{pmatrix} and \sum=\begin{pmatrix} \begin{matrix} 2&2 \\ 2& 1\end{matrix} & \begin{matrix} 3&0 \\ 2& 0\end{matrix} \\ \begin{matrix} 3&2 \\ 0& 0\end{matrix} & \begin{matrix} 2&1 \\ 1& 1\end{matrix} \end{pmatrix}. Find the 

E(X^{(2)}|X^{(1)}=x^{(1)}) and Cov \left ( X^{(2)}|X^{(1)}=x^{(1)} \right ).

02 Apr 2024
Answer :
Word Count : 808

To find \( E(X^{(2)}|X^{(1)}=x^{(1)}) \) and \( \text{Cov} \left ( X^{(2)}|X^{(1)}=x^{(1)} \right ) \), we'll use the properties of conditional expectation and covariance.

Given that \( X^{(1)} = x^{(1)} \), we know that \( X^{(2)} \) follows a conditional normal distribution. The conditional expectation \( E(X^{(2)}|X^{(1)}=x^{(1)}) \) is the mean of this conditional normal distribution, and the conditional covariance \( \text{Cov} \left ( X^{(2)}|X^{(1)}=x^{(1)} \right ) \) is the covariance matrix.

First, let's write down the properties of conditional normal distributions:

1. Conditional Mean:
\[ E(X^{(2)}|X^{(1)}=x^{(1)}) = \mu_2 + \Sigma_{21}\Sigma_{11}^{-1}(x^{(1)} - \mu_1) \]

2. Conditional Covariance:
\[ \text{Cov} \left ( X^{(2)}|X^{(1)}=x^{(1)} \right ) = \Sigma_{22} - \Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12} \]

Given:
\[ \mu = \begin{pmatrix} -4\\ \frac{1}{4} \\ 0 \end{pmatrix} \]
and
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Related Questions

Let X\sim N_4\left ( \mu ,\sum \right ), where \mu =\begin{pmatrix} 3\\ -2 \\1 \\ -2 \end{pmatrix}  and \sum=\begin{pmatrix} 4 & 0 & 0 & 0\\0 & 3 & 0 &0 \\ 0 & 0 & 2 & -2\\ 0 & 0 & -2 & 5 \end{pmatrix}. Check the independence of the (i) X_2 and X_1 (ii) (X_2 ,X_4  ) and ( X_1 , X_3 ) (iii) ( X_1 , X_2 ) and ( X_3 , X_4 ).

X=\binom{X^{(1)}}{X_{(2)}}\sim N_4\left ( \mu ,\sum \right ),  where \begin{pmatrix} -4\\ \frac{1}{4} \\ 0 \end{pmatrix} and \sum=\begin{pmatrix} \begin{matrix} 2&2 \\ 2& 1\end{matrix} & \begin{matrix} 3&0 \\ 2& 0\end{matrix} \\ \begin{matrix} 3&2 \\ 0& 0\end{matrix} & \begin{matrix} 2&1 \\ 1& 1\end{matrix} \end{pmatrix}. Find the 

E(X^{(2)}|X^{(1)}=x^{(1)}) and Cov \left ( X^{(2)}|X^{(1)}=x^{(1)} \right ).

Obtain the maximum likelihood estimator of the mean vector and variance-covariance matrix of the multivariate normal distribution.

Let\underset{\sim }{X} =\binom{X_1}{X_2}  has the following joint density function

f\left ( X_1,X_2 \right )=\left\{\begin{matrix} 4x_1x_2 &,0<x_1<1,0<x_2<1, \\ 0 & ,\textup{otherwise.} \end{matrix}\right.

Find the marginal distributions, mean vector and variance-covariance matrix. Also, comment on the independence of X_1 and X_2

State whether the following statements are true or false and also give the reason in support of your answer 

(a) The covariance matrix of random vectors \underset{\sim }{X} and  \underset{\sim }{Y}is symmetric.

(b) If \underset{\sim }{X} is a p-variate normal random vector, then every linear combination   \underset{\sim }{C{}'}\underset{\sim }{X},where  \underset{\sim }{C}_{P\times 1}is a scalar vector, is also p-variate normal vector.

(c) The trace of matrix \begin{pmatrix} 3 & -2\\ -2 &6 \end{pmatrix}   is 9.

(d) If a matrix is positive definite then its inverse is also positive definite.

(e) If \underset{\sim }{X} -N_2 \left ( \binom{2}{1},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right )  and   

 \underset{\sim }{Y} -N_2 \left ( \binom{-1}{3},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right ),  then

\underset{\sim }{X}+\underset{\sim }{Y} -N_2  \left ( \binom{1}{1},\begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \right )

Consider the following data of 11 samples on 8 variables by Anscombe, Francis J. (1973):

x_1 x_2 x_3 x_4 y_1 y_2 y_3 y_4
10 10 10 8 8.04 9.14 7.46 6.58
8 8 8 8 6.95 8.14 6.77 5.76
13 13 13 8 7.58 8.74 12.74 7.71
9 9 9 8 8.81 8.77 7.11 8.84
11 11 11 8 8.33 9.26 7.81 8.47
14 14 14 8 9.96 8.10 8.84 7.04
6 6 6 8 7.24 6.13 6.08 5.25
4 4 4 19 4.26 3.10 5.39 12.50
12 12 12 8 10.84 9.13 8.15 5.56
7 7 7 8 4.82 7.26 6.42 7.91
5 5 5 8 5.68 4.74 5.73  

If the vector x=\begin{pmatrix} x_1\\ x_2 \\x_3 \\ x_4 \end{pmatrix}  and y=\begin{pmatrix} y_1\\y_2 \\ y_3 \\ y_4 \end{pmatrix}  then obtain the sample covariance matrix between x and y

Source: Anscombe, Francis J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17– 21. doi: 10.2307/2682899.

 Define the following:
(i) Covariance Matrix
(ii) Mahalanobis D2
(iii) Hotelling’s T2
(iv) Clustering
(v) Relationship between (ii) and (iii). 

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