Probability and Mathematical Statistics

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Explain the concept of likelihood and derive the likelihood and associated functions of interest for simple models
• Explain the concepts of analysis of variance and use them to investigate factorial dependence
• Construct confidence intervals for unknown parameters.
• Test statistical hypotheses.
• State the central limit theorem, and apply it.
• Analyse the dependence of a binary response variable on explanatory variables using logistic regression models
• Define basic discrete and continuous distributions, be able to apply them and simulate them in simple cases
• Explain the concepts of probability, including conditional probability
• Summarise the main features of a data set (exploratory data analysis).
• Explain the concepts of random sampling, statistical inference and sampling distribution, and state and use basic sampling distributions.
• Describe the main methods of estimation and the main properties of estimators, and apply them
• Explain the concepts of a compound distribution, and apply them.
• Explain the concepts of random variable, probability distribution, distribution function, expected value, variance and higher moments, and calculate expected values and probabilities associated with the distributions of random variables
• Explain the concepts of independence, jointly distributed random variables and conditional distributions, and use generating functions to establish the distribution of linear combinations of independent random variables.
• Investigate relationships between variables using regression models.
• Use appropriate software (say R) to fit a multiple linear regression model to a data set and interpret the output.
• Define a probability generating function, a moment generating function, a cumulant generating function and cumulants, derive them in simple cases, and use them to evaluate moments
• Explain the principles of Bayesian statistical inference
• Recall the definition of a generalised linear model
• Recall the principles of statistical model selection, and apply a range of different approaches to model selection

Probability theory:

• Basic concepts; Axioms; Addition laws; Independence
• Random variables and probability distributions; Expectation, moments and variance;
• Distribution function
• Discrete and continuous distributions; Calculating probabilities and expectations;
• Transformations of random variables
• Examples: binomial, Poisson, exponential, normal etc.
• Generating functions; Properties and applications; Cumulants
• Joint distributions; Independence; Covariance and correlation
• Conditional probability; Bayes theorem; Compound distributions

Statistical inference:

• Sampling concepts; Samples and populations; Sampling distribution
• The central limit theorem; normal approximations
• Point estimation; Efficiency; Bias; Consistency; Mean squared error; Method of Moments
• Confidence intervals; Normal, Poisson and binomial examples
• Hypothesis testing; Terminology; Normal, Poisson and binomial examples; Goodness-of-fit tests
• Likelihood; Maximum likelihood estimation and its asymptotic properties
• Introduction to Bayesian ideas; Prior and posterior distributions

Statistical modelling:

• Summarising data; Summary statistics; Simple graphical displays
• Regression and linear models; Least squares estimation; Inference for regression coefficients;
• Comparing models; Multiple regression; Residuals and model criticism; Prediction
• Analysis of variance
• Use appropriate software (say R) to fit a multiple linear regression model to a data set and interpret the output.
• Introduction to generalised linear models; Exponential family; Logistic regression
• Concepts and methods of statistical model selection; Hypothesis testing approaches;
• Information criteria.

Lectures, assigned problems, private study

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.