Statistical Astronomy Lectures

Astronomy 3/4H: SA-I

Statistical Astronomy I

Dr M. Hendry, Room 312 Kelvin Building

10 lectures, starting 11th October 2000

Course Content (10 lectures)

Section 1: Mathematical Building Blocks

    1.1:    The theory of probability, Probability as the relative frequency of outcomes. Law of Addition. Conditional Probability. Law of multiplication
    1.2:    Statistical independence
    1.3:    Probability distributions. Discrete random variables; Poisson distribution. Continuous random variables. Cumulative distribution function.
                Uniform and normal distributions
    1.4:    Expectation and other measures of a distribution. Expected, or mean, value. Median. Mode. Variance. Skewness. Kurtosis. Variance of a
                function of random variable
    1.5:    Variable transformations
    1.6:    Probability integral transform
    1.7:    Multivariate distributions. Joint PDF of two or more random variables. Marginal distributions. Conditional distributions. Bayes' theorem
    1.8:    Statistical independence revisited
    1.9:    The bivariate normal distribution

Section 2: Statistical Building Blocks

    2.1:    The sampling distribution
    2.2:    Parameter estimation. Definition of a statistic. Estimators. Bias of an estimator. Risk of an estimator. The sample mean. The law of large
                numbers. The central limit theorem
    2.3:    The principle of maximum likelihood. Likelihood functions. Maximum likelihood estimators
    2.4:    The principle of least squares. Least squares fits to straight lines. Linear regression
    2.5:    Goodness of fit. The chi-squared distribution
    2.6:    Fitting general models

Sec. 3: Hypothesis Tests

    3.1:    Simple hypothesis tests. Test statistics. Null and alternative hypotheses. Critical regions
    3.2:    Incorrect decisions. Type I and II errors. Power of a hypothesis test
    3.3:    Level of significance
    3.4:    Two-tailed tests
    3.5:    Goodness of fit for discrete distributions. The binomial distribution
    3.6:    Are two distributions equal?. The Kolmogorov-Smirnoff test
    3.7:    The student's t test
    3.8:    Testing the difference of means
    3.9:    Testing the ratio of variances
    3.10: Testing hypotheses on the correlation coefficient

Section 4: Point and Interval Estimation

4.1: Defining confidence intervals
4.2: Interpreting confidence intervals

Martin Hendry,
October 2000

Lecture Notes

Course Contents and Introduction

Mathematical Building Blocks: Section 1.1

Statistical Independence and Probability Distributions

Expectation and Other Measures of a Distribution

Variable Transformations, Multivariate Distributions, Statistical Independence

The Bivariate Normal Distribution

Statistical Building Blocks: Sampling Distribution to Central Limit Theorem

The Principles of Maximum Likelihood and Least Squares