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# ST111 and ST112: Probability A & B

###### Lecturer(s): Dr Jon Warren

Prerequisites: MA131 Analysis I, MA132 Foundations.

Leads to: ST104 Statistical laboratory, ST220 Introduction to Mathematical Statistics, ST202 Stochastic Processes, MA3H2 Markov processes and percolation theory, and to numerous statistical, probabilistic, operational research and econometric courses.

Commitment: This module runs in Term 2.

• ST111 - 15 hours of lectures, 2 tutorial hours (week 3 and week 5)
• ST112 - 15 hours of lectures, 2 tutorial hours (week 7 and week 9).

Aims: To lay the foundation for all subsequent modules in probability and statistics, by introducing the key notions of mathematical probability and developing the techniques for calculating with probabilities and expectations.

Content (part A):

1. Experiments with random outcomes: the notions of events and their probability. Operations with sets and their interpretation. The addition law and axiomatic definition of a probability space.
2. Simple examples of discrete probability spaces. Methods of counting: inclusion-exclusion formula and multinomial co-efficients. Examples including the birthday problem and coupon collecting.
3. Simple examples of continuous probability spaces. Points chosen uniformly at random in space.
4. Independence of events. Conditional probabilities. Simpson’s paradox. Bayes theorem.
5. Binomial probabilities. The law of large numbers, Poisson and Gaussian approximations and their applications.

Content (part B):

1. The notion of a random variable and its distribution. Examples in both discrete and continuous settings. Probability mass functions and density functions. Cumulative distribution functions.
2. Joint distributions. Independence of random variables.
3. Expectation of random variables. Properties of expectation.
4. Variance and Chebyshev's inequality. Covariance and the Cauchy-Schwartz inequality.
5. Addition of independent random variables: convolutions. Generating functions, Moment generating functions and their use to compute convolutions.
6. Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma and Gaussian. Their properties, genesis and inter-relationships.
7. The law of large numbers and the Central limit theorem.

Books:

• Durrett, Elementary Probability for Applications.
• Grimmett and Walsh, Probability- An Introduction.
• Grimmett and Stirzaker, One Thousand Exercises in Probability
• Sheldon Ross, A first course in Probability.

Assessment: 10% assessed work (during term 2) and 90% written examination (in term 3).

Deadlines:

• ST111 assignments are due on Tuesdays of weeks 4 and 6
• ST112 assignments are due on Tuesdays of weeks 8 and 10

Feedback: Feedback on your assignments will be given within 2 weeks of submission