Draft syllabus for a 10-hour graduate short course in Probability and Statistics for Earth and Planetary Scientists

P.B. Stark.

Last edited 19 April 2015

Lecture 1: Probability

1. Counting and combinatorics
   • Sets: unions, intersections, partitions
   • De Morgan's Laws
   • The Inclusion-Exclusion principle
   • The Fundamental Rule of Counting
   • Combinations
   • Permutations
   • Strategies for counting
2. Theories of probability
   • Equally likely outcomes
   • Frequency Theory
   • Subjective Theory
   • Shortcomings of the theories
   • Rates versus probabilities
   • Measurement error
   • Where does probability come from in physical problems?
   • Making sense of geophysical probabilities
     • Earthquake probabilities
     • Probability of magnetic reversals
     • Probability that Earth is more than 5B years old

Lecture 2: Probability, continued

3. Axiomatic Probability
   • Outcome space and events, events as sets
   • Kolmogorov's axioms (finite and countable)
   • Analogies between probability and area or mass
   • Consequences of the axioms
     • Probabilities of unions and intersections
     • Bounds on probabilities
     • Bonferroni's inequality
     • The inclusion-exclusion rule for probabilities
   • Conditional probability
     • The Multiplication Rule
     • Independence
     • Bayes Rule
4. Random variables.
   • Probability distributions of real-valued random variables
   • Cumulative distribution functions
   • Discrete random variables
     • Probability mass functions
     • The uniform distribution on a finite set
     • Bernoulli random variables
     • Random variables derived from the Bernoulli
       • Binomial random variables
       • Geometric
       • Negative binomial
     • Hypergeometric random variables
     • Poisson random variables: countably infinite outcome spaces

Lecture 3: Random variables, contd.

5. Random variables, continued
   • Continuous and "mixed" random variables
   • Probability densities
     • The uniform distribution on an interval
     • The Gaussian distribution
   • The CDF of discrete, continuous, and mixed distributions
   • Distribution of measurement errors
     • The box model for random error
     • Systematic and stochastic error
6. Independence of random variables
   • Events derived from random variables
   • Definitions of independence
   • Independence and "informativeness"
   • Examples of independent and dependent random variables
   • IID random variables
   • Exchangeability of random variables
7. Marginal distributions
8. Point processes
   • Poisson processes
     • Homogeneous and inhomogeneous Poisson processes
     • Spatially heterogeneous, temporally homogenous Poisson processes as a model for seismicity
     • The conditional distribution of Poisson processes given N
   • Marked point processes
   • Inter-arrival times and inter-arrival distributions
   • Branching processes
     • ETAS

Lecture 4: Expectation and Simulation

9. Expectation
   • The Law of Large Numbers
   • The Expected Value
     • Expected value of a discrete univariate distribution
       • Special cases: Bernoulli, Binomial, Geometric, Hypergeometric, Poisson
     • Expected value of a continuous univariate distribution
       • Special cases: uniform, exponential, normal
     • Expected value of a multivariate distribution
   • Standard Error and Variance.
     • Discrete examples
     • Continuous examples
     • The square-root law
     • Standardization and Studentization
     • The Central Limit Theorem
   • The tail-sum formula for the expected value
   • Conditional expectation
     • The conditional expectation is a random variable
     • The expectation of the conditional expectation is the unconditional expectation
   • Useful probability inequalities
     • Markov's Inequality
     • Chebychev's Inequality
     • Hoeffding's Inequality
     • Jensen's inequality
10. Simulation
    • Pseudo-random number generation
      • Importance of the PRNG. Period, DIEHARD
    • Assumptions
    • Uncertainties
    • Sampling distributions

Lecture 5: Testing

11. Hypothesis tests
    • Null and alternative hypotheses, "omnibus" hypotheses
    • Type I and Type II errors
    • Significance level and power
    • Approximate, exact, and conservative tests
    • Families of tests
    • P-values
      • Estimating P-values by simulation
    • Test statistics
      • Selecting a test statistic
      • The null distribution of a test statistic
      • One-sided and two-sided tests
    • Null hypotheses involving actual, hypothetical, and counterfactual randomness
    • Multiplicity
      • Per-comparison error rate (PCER)
      • Familywise error rate (FWER)
      • The False Discovery Rate (FDR)

Lecture 6: Tests and Confidence sets

12. Tests, continued
    • Parametric and nonparametric tests
      • The Kolmogorov-Smirnov test and the MDKW inequality
      • Example: Testing for uniformity
      • Conditional test for Poisson behavior
    • Permutation and randomization tests
      • Invariances of distributions
      • Exchangeability
      • The permutation distribution of test statistics
      • Approximating permutation distributions by simulation
      • The two-sample problem
    • Testing when there are nuisance parameters
13. Confidence sets
    • Definition
    • Interpretation
    • Duality between hypothesis tests and confidence sets
    • Tests and confidence sets for Binomial p
    • Pivoting
      • Confidence sets for a normal mean
        • known variance
        • unknown variance; Student's t distribution
    • Approximate confidence intervals using the normal approximation
      • Empirical coverage
      • Failures
    • Nonparametric confidence bounds for the mean of a nonnegative population
    • Multiplicity
      • Simultaneous coverage
      • Selective coverage