A list of likely exam questions.


  1. Discuss the difference between the definition of probability in probability space and empirical probability.
  2. "Prove" Bayes' theorem and discuss its relevance in inference problems.
  3. Describe the concept of convergence in probability theory.
  4. Prove Chebyshev's inequality and use it to find the weak law of large numbers.
  5. Describe the link between probability generating functions and momenta of discrete distributions.
  6. Find the probability generating function of a sum of random variables.
  7. Find the probability generating function of common distributions, Poisson, Bernoulli, Binomial.
  8. Find the characteristic function of the Gaussian distribution.
  9. Give an outline of the proof of the Central Limit Theorem.
  10. Describe the geometric distribution.
  11. Describe the negative binomial distribution.
  12. Describe the hypergeometric distribution.
  13. Describe the elementary model of cancer onset vs. age.
  14. Find the formula for the detected rate vs. true rate for paralizable and non-paralizable counters.
  15. Prove the formula for the coincidence rate of multiple detectors.
  16. Find the high-rate limit of the Poisson distribution.
  17. Describe in detail the multivariate Gaussian distribution.
  18. Describe the gamma distribution and prove the formulas for expectation value and variance.
  19. Demonstrate the formula for the distribution of distance to the nearest neighbor (in the limit of large number density).
  20. Repeat the proof that the correlation coefficient is lies in the (-1,+1) interval.
  21. Find the probability distribution of the sample mean.
  22. Explain order statistics.
  23. What is a box plot?
  24. What is a kernel density estimate?
  25. What is an empirical probability density?
  26. Discuss the algorithmic generation of pseudo-random numbers.
  27. Describe the transformation method to generate arbitrarily distributed pseudo-random numbers
  28. Describe the acceptance-rejection method and provide an elementary proof of its validity.
  29. Describe the statistical bootstrap.
  30. Describe the principle of maximum likelihood (MaxL).
  31. Discuss the application of MaxL to normally distributed data, to estimate mean and/or standard deviation.
  32. Discuss the application of MaxL to data extracted from a Poisson distribution to estimate the mean.
  33. Prove the Bartlett identities.
  34. The Cramer-Rao-Fisher bound and the Fisher information.
  35. Define the Shannon information.
  36. How to use the likelihood to estimate the confidence intervals of model parameters.
  37. Describe the method of least squares.
  38. Describe linear regression.
  39. How to use a statistic to discriminate between null and alternative hypothesis.
  40. Prove the Neyman-Pearson lemma.
  41. What are p-values and how do you use them in multiple tests of the null hypothesis?
  42. Describe Wilks' theorem.
  43. Describe the Naive Bayesian Classification.
  44. Describe the k-means algorithm.
  45. Describe the principal component method.




December 2023.