Indicative Module Content:

• Revision on Probability:
Probability
Conditional probability
Independence
Law of total probability
Random Variables
Cumulative Distribution Function
Density and Mass Functions
Independent and Identically Distributed
Expectation
Moments
Moment Generating Functions
Bernoulli distribution
Binomial distribution
Geometric distribution
Negative binomial distribution
Poisson distribution
Uniform distribution
Gamma distribution
Normal (or Gaussian) distribution

• Multivariate Probability Distribution
Random vectors
Discrete random vectors
marginal probability mass functions
Continuous random vectors
joint probability density function
Conditional Distributions
conditional probability mass function
conditional probability density function
Independence
Conditional Expectation
Properties of the Expectation Operator

• Some more basic concepts
Covariance
correlation
Jointly Gaussian random variables
Bivariate normal distribution
Mutually Independent Random Variables

• Random samples
Statistics and Estimators
Sample mean and sample variance, and their properties
The chi-squared distribution
Student's t distribution
F distribution

• Convergence concepts
Chebyshev's Inequality
Convergence in probability
Convergence in distribution
Weak law of large numbers
Consistent estimators
Central Limit Theorem

• Parametric statistical inference, point estimation methods
Method of moments
Maximum Likelihood Estimation
Iterative likelihood maximization
Newton-Raphson scheme

• Properties of estimators: how to compare estimators
Unbiased/biased
Consistent estimators
Efficiency
Cramer-Rao lower bound
Sufficient statistics
Fisher-Neyman factorization
Rao-Blackwell theorem
Exponential families

• Confidence intervals
The pivot method
Approximate confidence intervals
Newton-Raphson

• Hypothesis testing I
Standard material
one-sided, two-sided, and composite tests

• Hypothesis testing II
Neyman-Pearson lemma
Significance level and power
Uniformly most powerful tests
Generalized likelihood ratio tests
Wilks Thereom

• Introduction to Basian inference (only one week on this)
Bayesian inference
Prior distributions
Posterior distributions
Conjugacy

Student Effort Type	Hours
Lectures	0
Total	0

Not applicable to this module.

Not applicable to this module.
 

Assessment Strategy  

Description	Timing		Component Scale		% of Final Grade
Examination: Assessment will be based on the midterm and final exams.	Unspecified	No	Alternative linear conversion grade scale 40%	No	100

 

Remediation Type	Remediation Timing
In-Module Resit	Prior to relevant Programme Exam Board

• Feedback individually to students, post-assessment

Not yet recorded.

Main reference:
Robert Hogg, Joseph McKean, and Allen Craig: "Introduction to Mathematical Statistics", 8th edition (2018).

Further reading:
1) Morris H. DeGroot and Mark J. Schervish: "Probability and Statistics", 4th edition (2010).
2) George Casella and Roger L. Berger : "Statistical Inference", 2nd edition (2002).