ST115: Introduction to Probability
Lecturer(s)
Prerequisite(s): MA131 Analysis, MA132 Foundations,
Commitment: 3 lectures/week, 1 tutorial/fortnight, 1 exercise class/week. This module runs in Term 2.
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:
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1. |
Experiments with random outcomes: the notions of events and their probability. Operations with sets and their interpretation. The addition law. |
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2. |
Simple examples of discrete probability spaces. Methods of counting: inclusion-exclusion formula and Binomial co-efficients. |
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3. |
Simple examples of continuous probability spaces. Points chosen uniformly at random in space. |
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4. |
Independence of events. Conditional probabilities. Bayes theorem. |
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5. |
The notion of a random variable. Examples in both discrete and continuous settings. Indicator random variables. |
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6. |
The notion of the distribution of a random variable. Probability mass functions and density functions. Cumulative distribution functions. |
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7. |
Expectation of random variables. Properties of expectation. |
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8. |
Mean and variance of distributions. Chebyshev's inequality. |
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9. |
Independence of random variables. Joint and conditional distributions. Covariance. Cauchy-Schwartz inequality. |
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10. |
Addition of independent random variables: convolutions. Generating function and use to compute convolutions. |
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11. |
Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma and Gaussian. Their properties, genesis and inter-relationships.
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Then the following topics to be covered in second year may be introduced at the end of this module.
Sequences of random variables.
Convergence in probability and distribution. Examples including the
Weak law of large numbers.
Generating functions and relationship with with convergence.
Central limit theorem.
Books:
Ross, A first course in probability, Prentice Hall, 1994
Pitman, Probability, Springer texts in Statistics
Suhov and Kelbert, Probability and Statistics by Example: Basic probability and Statistics.
Assessment: 100% by 2 hr examination.
Resources for Current Students (restricted access)
