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# Department of Mathematical Sciences

# MATH1597 Probability I

## Michaelmas Term

Probability is a concept with applications in all numerate disciplines e.g. in mathematics, science and technology, medicine, engineering, agriculture, economics and many other fields. In this course, the theory of probability is developed with the calculus and analysis available and with applications in mind. Among the topics covered are: probability axioms, conditional probability, special distributions, random variables, expectations, generating functions, applications of probability, laws of large numbers, central limit theorems.

## Outline of course

Aim: To develop probabilistic insight and computational skills.

• Introduction to probability: chance experiments, sample spaces,events, assigning probabilities. Probability axioms and interpretations.
• Conditional probability: theorem of total probability, Bayes theorem,independent events. Applications of probability.
• Random variables: discrete probability distributions and distribution functions, binomial, Poisson, Poisson approximation to binomial, transformations of random variables.
• Continuous random variables: probability density functions, normal distribution, normal approximation to binomial.
• Joint, marginal and conditional distributions.
• Expectations: expectation of transformations, variance, covariance, expectations of expectations, Chebyshev's inequality, weak law of large numbers. Moment-generating functions.
• Central-limit theorems.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.