Question

Suppose the random variables X1 ,X2 and X3have the covariance matrix

                        \sum =\begin{bmatrix} 1 &-1 &0 \\ -1& 5& 0\\ 0& 0 & 2 \end{bmatrix}

Find all principal components.

07 Feb 2021
Answer :
Word Count : 668
We are asked to find the principal components of a random vector $(X_1, X_2, X_3)$ with covariance matrix $$ \Sigma = \begin{bmatrix} 1 & -1 & 0 \\ -1 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix}. $$ Let’s solve this step by step, manually. --- ### Step 1: Find eigenvalues Principal components are obtained from the eigenvalues and eigenvectors of the covariance matrix $\Sigma$. The characteristic equation is: $$ \det(\Sigma - \lambda I) = 0. $$ So we compute: $$ \Sigma - \lambda I = \begin{bmatrix} 1-\lambda & -1 & 0 \\ -1 & 5-\lambda & 0 \\ 0 & 0 & 2-\lambda \end{bmatrix}. $$ The determinant is: $$ \det(\Sigma - \lambda I) = \begin{vmatrix} 1-\lambda & -1 & 0 \\ -1 & 5-\lambda & 0 \\ 0 & 0 & 2-\lambda \end{vmatrix}. $$ Since the matrix is block-diagonal in the 3rd variable, we can write: $$ \det(\Sigma - \lambda I) = (2 - \lambda) \cdot \det \begin{bmatrix} 1-\lambda & -1 \\ -1 __________ ____ ____ __________ ____ ______ ________ _________.
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Suppose that people arrive at a bus stop in accordance with a Poisson process with rate \lambda . The bus departs at time t . Let X denote the total amount of waiting time of all those that get on the bust at time t . We want to determine Var ( X ) . Let N (t ) denote the number of arrivals by time t

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ii) Argue that Var  [ x \mid N (t) ] = N (t) t^{2} /12

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Let  {X}' = [ X_{1} , X_{2}] be a random vector with mean vector { \mu }'_{x} =[ \mu_{1} ,\mu_{2} ]  and variance-covariance matrix

                                                    \sum _{x} = \begin{bmatrix} \sigma _{11} &\sigma_{12} \\ \sigma_{12} & \sigma _{22} \end{bmatrix}

Find the means and covariance matrix for the linear combinations 

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                                                          Z_{2} = X_{1} + X_{2}

or  

           Z = \begin{bmatrix} Z_{1} \\Z _{2} \end{bmatrix} =\begin{bmatrix} 1 &-1 \\ 1 &1 \end{bmatrix}\begin{bmatrix} X_{1}\\ X_{2} \end{bmatrix} = CX

       in terms of \mu _{x} and\sum _{X}

Suppose the joint density function of X and Y is given by

f ( x, y) = \left\{\begin{matrix} 4y\: (x-y) e^{-(x+y)} \: \: \: \: \: \: \: \: \: 0< x < \infty ,0\leq y \leq x & \\ 0, \; \; \; \; \; \; \; \; \; \; \; \; otherwise& \end{matrix}\right.

 compute\; E \left [ X \mid Y =y \right ]

The joint probability mass function of X and Y , p ( x , y ) , is given by

 p( 1,1)= \frac{1}{4} ,\: \: \: \: p(2,1)= \frac{1}{4} ,\: \: \: \: p( 3,1)= \frac{1}{16}

p( 1,2)= \frac{1}{16} ,\: \: \: \: p(2,1)= 0 ,\: \: \: \: p( 3,2)= \frac{1}{16}

p( 1,3)= 0 ,\: \: \: \: p(2,3)=\frac{1}{16} ,\: \: \: \: p( 3,3)= \frac{1}{4}

i)  Computer E [ X \mid Y = i ] for\: \: i = 1,2,3

ii)      Find P[x/y=1]

iii)   Marginal distributions of X and Y

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            R = \begin{bmatrix} Y & Z_1 &.Z_2 \\ 1.0 & -.35 &.82 \\ -.35 &1.0 & -.60\\ .82&-.60 & 1.0 \end{bmatrix}

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ii) Calculate the partial correlation coefficient  r_{yz_{1}} z_{2} and interpret this quantity with respect to the interpretation provided for  r_{yz_{1}} in Part (i)

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\sum =\begin{bmatrix} 16 &-2 &4 \\ -2& 9&1 \\ 4&1 & 25 \end{bmatrix}

i) Determine p and V 1/2

ii) Multiply your matrices to check the relation  V 1/2 P V 1/2

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