# 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.

# REFERENCES AND FURTHER READING

Mardia KV, Kent JT, Bibby JM. (1976). *Multivariate Statistics.* London U.K: Academic Press.

Pritchard JK, Stephens M, Donnelly P. (2000). Inference of population structure using multilocus genotype data. *Genetics* 155(2):945-959.