Credits: 10 |
Convenor: Mr. B. English |
Semester: 1 (weeks 7 to 12) |

Prerequisites: |
essential: MC160 | desirable: MC265 |

Assessment: |
Coursework: 20% | One and a half hour examination: 80% |

Lectures: |
18 | Classes: |
5 |

Tutorials: |
none | Private Study: |
47 |

Labs: |
5 | Seminars: |
none |

Project: |
none | Other: |
none |

Total: |
75 |

- be able to compute moment generating functions and probability generating functions and exploit these to obtain moments and the distributions of functions of random variables, and limiting distributions;
- know what is meant by a joint probability distribution or probability density function, a marginal distribution and a conditional distribution;
- be able to construct a joint distribution in simple examples;
- know how to obtain marginal and conditional distributions from joint distributions;
- be able to find expectations, variances, covariances, correlations, regressions and conditional variances from joint distributions, and understand their significance;
- know how to determine the distribution (or joint distribution) of functions of one or more random variables using appropriate techniques, which include the use of moment generating functions, probability generating functions, the distribution function, and Integral Transformation Theorem.

- Maintain and extend students basic calculus skills including series expansion and summation, and univariate and bivariate integration.
- Provide further essential probabilistic concepts and mathematical techniques for later modules in probability and statistics.

Multivariate random variables (mainly restricted to bivariate random variables). Joint distributions; marginal and conditional distributions; independence. Expectations, conditional expectations and regression; covariance and the coefficient of correlation. Theorems: Unconditional means and variance from conditional means and variances; linear regression and the coefficient of correlation. Introduction to bivariate integrals; bivariate continuous random variables, joint probability density functions; marginal and conditional distributions. Joint moment and probability generating functions. The bivariate normal distribution and its basic properties.

Functions of random variables. Functions of a single random variable. Use of the
moment and probability generating functions to find the distribution
of functions of random variables; examples including the derivation of the
distribution, the sum of independent Poisson variables, and the Central
Limit Theorem. Functions of
continuous random variables via the distribution function; the distribution of
sums, ratios and products of random variables; examples include the
derivation of the *F* distribution.
The distribution of functions of random variables via the Integral Transformation
Theorem; examples to include the distribution of linear combinations of
independent normal random variables.

**M. H. DeGroot**,
* Probability and Statistics, 2nd edition*,
Addison-Wesley, 1986.

**J. E. Freund and R. E. Walpole**,
*Mathematical Statistics, 3rd edition*,
Prentice-Hall.

**W. Mendenhall, R. L. Scheaffer and D. D. Wackerly**,
*Mathematical Statistics with Applications, 4th edition*,
Duxbury Press, 1990.