# EXERCISES

• 1. What is the most probable value under a univariate Gaussian distribution? What is its probability?
• 2. Use the joint probability rule to argue that a multivariate Gaussian with diagonal covariance is nothing but the product of univariate Gaussians.
• 3. Show that the average is also the MLE for the parameter of the Poisson distribution. Explain why this is consistent with what I said about the average of the Gaussian distribution in Chapter 1.
• 4. Fill in the components of the vectors and matrices for the part of the multivariate Gaussian distribution:

• 5. Derive the MLE for the covariance matrix of the multivariate Gaussian (use the matrix calculus tricks I mentioned in the text).
• 6. Why did we need Lagrange multipliers for the multinomial MLEs, but not for the Guassian MLEs?
• 7. Notice that I left out the terms involving factorials from the multinomial distribution when I calculated the LRT. Show/explain why these terms wonâ€™t end up contributing to the statistic.