1PS2: Probability and Statistics
Additional task: Moment-generating functions
Students taking this module at ‘Intermediate Level’ (Level I) are required to complete an additional task for the module. This will be a report written on the topic of moment-generating functions. The report should give an introduction to the topic, accessible to a student who has taken 1PS.
- The additional task is intended to take 7-10 hours to complete.
- The additional task will draw upon material introduced later in the module, but you can begin to work on the task now.
- Your submission for the additional task should be a pdf file, which could be handwritten or scanned or typed in LATEX .
- Your submission for the additional task should be 3–4 pages of A4 if handwritten.
- The additional task is coursework as opposed to an exam, and you should feel free to discuss it with each other, or to discuss it with me (perhaps during my Office Hours).
- This additional task will be graded and will contribute 10% to your final grade for 1PS2 Probability and Statistics. Further details on assessment arrangements can be found at Assessment and Feedback.
- The submission deadline for the additional task is 17:00 on 24 April 2024.Your task must be uploaded to Additional Task Assignment on the 1PS(2) canvas course.
Task details: During the module we have seen that mass, density and distribution functions store key information about random variables. There are also several other interesting ways to represent this information and for this additional task you should write a report giving an introduction to moment-generating functions.
Your report should include:
- the definition of the moment-generating function of a random variable;
- calculations which show how to find the moment-generating functions for some simple random variables;
- calculations of the moment-generating functions of some common types of random variables we have met through the course (binomial, Poisson, geometric, exponential, ...);
- a discussion of some useful properties of moment-generating functions.
- the moment-generating function of the sum of independent random variables;
- use of the moment-generating functions to prove Chernoff’s inequality;
- a comparison of moment-generating functions with other functions, such as probability-generating functions or characteristic functions.
For a higher first-class mark you should research and report on at least oneextra topic beyond those discussed above.
Grading: The grading will take the following factors into account:
- The correctness and the quality of explanation of the mathematics.
- The level of the mathematics included.
- The amount of independent research demonstrated.
- The organisation, development and presentation of the report.
You should research into this topic yourself and present your findings so thatthe content is accessible to a diligent student who has taken 1PS.NB! Plagiarism is a serious offence. You should properly cite any referencesused, with Harvard referencing, and formulate all ideas in your own words.