Question

(a)(30%) Given $\Sigma = \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix}$ Find $|\Sigma|$, which is notation for determinant of $\Sigma$ and $\Sigma^{-1}$, inverse matrix of $\Sigma$ Note that with these results you can easily show Equation 5.11 Text $P(x_1, x_2) = \frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \exp \left[ -\frac{1}{2(1 - \rho^2)} (z_1^2 - 2 \rho z_1 z_2 + z_2^2) \right]$

          (a)(30%) Given
$\Sigma = \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix}$
Find $|\Sigma|$, which is notation for determinant of $\Sigma$ and $\Sigma^{-1}$, inverse matrix of $\Sigma$
Note that with these results you can easily show Equation 5.11
Text
$P(x_1, x_2) = \frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1 - \rho^2}} \exp \left[ -\frac{1}{2(1 - \rho^2)} (z_1^2 - 2 \rho z_1 z_2 + z_2^2) \right]$

(a)(30%) Given
Σ =
< b m a t r i x >
Find |Σ|, which is notation for determinant of Σ and Σ^-1, inverse matrix of Σ
Note that with these results you can easily show Equation 5.11
Text
P(x1, x2) = (1)/(2 πσ1 σ2 √(1 - ρ^2))[ -(1)/(2(1 - ρ^2)) (z1^2 - 2 ρ z1 z2 + z2^2) ]

Added by Monica W.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Hoan Nguyen

Step 1

Step 1: Write the covariance matrix $\Sigma=\begin{bmatrix}\sigma_1^2 & \rho\,\sigma_1\sigma_2\\[4pt]\rho\,\sigma_1\sigma_2 & \sigma_2^2\end{bmatrix}$. Show more…

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(a)(30%) Given $$ \Xi = \begin{bmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{bmatrix} $$ Find $|\Xi|$, which is notation for determinant of $\Xi$ and $\Xi^{-1}$, inverse matrix of $\Xi$ Note that with these results you can easily show Equation 5.11 Text $$ P(x_1,x_2) = \frac{1}{2 \pi \sigma_1 \sigma_2 \sqrt{1-\rho^2}} \exp \left[ -\frac{1}{2(1-\rho^2)} (z_1^2 - 2 \rho z_1 z_2 + z_2^2) \right] $$