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 won’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.

 
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