For a multivariate linear regression problem with multivariate output , the loss function is minimised with respect to the weight matrix by setting the partial derivative . Which of the following terms is used to denote the partial derivative matrix, in such a scenario?

For a multivariate linear regression problem, the loss function is minimized with respect to the weight vector by setting the partial derivative . Which of the following correctly shows the expansion of the partial derivative, ?

In the closed form solution to multivariate linear regression, why do we use the pseudo-inverse of instead of the simple inverse?

An engineer has IID Gaussian observations where both and are unknown. The engineer makes the following argument:

"The standardised sample mean is standard normal. Therefore, I can replace the unknown by the sample standard deviation and obtain the exact confidence interval Afterall, since the chi-squared confidence interval for is exact for a Gaussian population, this substitution must also be exact."

Which assessment is correct?

This question requires you to write your response on an answer sheet provided by the teaching team. Write your student ID on your answer sheet.

A power-electronics cooling fan is assumed to have an exponentially distributed operating lifetime

, measured in hours, with PDF

(a) Determine the rate parameter , including its units.

(b) Determine the variance and standard deviation of .

(c) An engineer claims that because the mean lifetime is 2000 h, most fans should fail close to 2000 h. Briefly explain why the exponential model does not support this interpretation.

Show all relevant working. (Note: You do

Hand your answer sheet to the teaching team at the end of the mid-semester test.