Question

The covariance matrix, \( \Sigma \), for the planar coordinates of a surveyed point is given as:

\[ \Sigma = \begin{bmatrix} 25 & 0.500 \\ 0.500 & 100 \end{bmatrix} \ \text{(in mm}^2\text{)} \]

The coefficient of correlation is \underline{\hspace{1cm}} (rounded off to 2 decimal places).

Hint:

The correlation coefficient between two variables from a covariance matrix is computed by dividing the covariance term by the product of standard deviations of the individual variables.

Answer 1

Correct Answer: 0.01

The coefficient of correlation \( \rho \) is calculated as: \[ \rho = \frac{{Cov}(X,Y)}{\sqrt{{Var}(X)} \cdot \sqrt{{Var}(Y)}} \] From the covariance matrix: \[ {Var}(X) = 25 { mm}^2,\quad {Var}(Y) = 100 { mm}^2,\quad {Cov}(X,Y) = 0.5 { mm}^2 \] \[ \rho = \frac{0.5}{\sqrt{25} \cdot \sqrt{100}} = \frac{0.5}{5 \cdot 10} = \frac{0.5}{50} = 0.01 \]