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Unit outline_

STAT2911: Probability and Statistical Models (Adv)

Overview

This unit is essentially an advanced version of STAT2011, with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common distributions including the Poisson, normal, beta and gamma families as well as the bivariate normal are introduced. Moment generating functions and convolution methods are used to understand the behaviour of sums of random variables. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The notions of conditional expectation and prediction will be covered as will be distributions related to the normal: chi^2, t and F. The unit has weekly computer classes where you will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 (MATH1X61 or MATH1971 or MATH1X21 or MATH1931 or MATH1X01 or MATH1906 or MATH1011) and a mark of 65 or greater in (DATA1X01 or MATH10X5 or MATH1905 or STAT1021 or ECMT1010 or BUSS1020 or MATH1X62 or MATH1972) None STAT2011 None Yes

Teaching staff

Coordinator Uri Keich, uri.keich@sydney.edu.au

Assessment

The census date for this unit availability is 2 April 2024
Type Description Weight Due Length
Supervised exam

Final exam
Final exam
60% Formal exam period 2 hours
Outcomes assessed:
Small test Quiz 1
Multiple choice and short answer questions
10% Week 05
Due date: 20 Mar 2024 at 11:00
50 mins
Outcomes assessed:
Assignment Assignment 1
Answers with calculations
2.5% Week 06
Due date: 29 Mar 2024 at 23:59
n/a
Outcomes assessed:
Small test Quiz 2
Multiple choice and short answer questions
10% Week 11
Due date: 08 May 2024 at 11:00
50 mins
Outcomes assessed:
Assignment Assignment 2
Answers with calculations
2.5% Week 12
Due date: 17 May 2024 at 23:59
n/a
Outcomes assessed:
Small test Computer exam
Computer test
10% Week 13
Due date: 20 May 2024 at 13:00
40 mins
Outcomes assessed:
Online task Computer work
Weekly lab assignments submitted at the end of the week.
4% Weekly n/a
Outcomes assessed:
Online task Weekly tutorial problems
Weekly tutorial problems submitted prior to the tutorial
1% Weekly N/A
Outcomes assessed:

Assessment summary

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Support for students

The Support for Students Policy 2023 reflects the University’s commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy 2023. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction and definitions; P(A U B), equiprobable spaces, birthday questions, conditional probability, independence, total probability law; Bayes Rule, Random Variables, pmf, Bernoulli, Binomial, Geometric, Negative Binomial; Lecture and tutorial (5 hr)
Week 02 Hypergeometric RV, Fisher's Exact Test, joint PMF, independent discrete RVs, deriving the Poisson distribution; Poisson approximation to the binomial distribution, faulty monitor, sum of independent RVs (convolution): binomial, Poisson RVs; Marginal conditional distribution (binomial, Poisson), Expectation (and L^1) definition and examples (Bernoulli, Poisson, binomial, geometric), Variance definition and calculation; Lecture and tutorial (5 hr)
Week 03 Variance calculation, Proof of claim on expectation of a function of a RV, variance of Bernoulli, Geometric and Poisson RVs, computing the expectation of a function of a random vector, L^1 is a vector space and expectation is a linear operator on it; Coupon collector problem; expectation of a product of ind RVs, L^2 is a vector subspace of L^1 corresponding to RVs with finite variance Covariance, Variance of a sum of RVs, Independence and 0 covariance, Variance of the binomial and hypergeometri Lecture and tutorial (5 hr)
Week 04 Variance of the negative binomial distribution, Markov and Chebyshev's Inequalities, Convergence in probability, Weak Law of Large Numbers, Multinomial distribution; Method of moments estimation: definition and examples, MLE: definition; MLE: Bernoulli, Poisson, Binomial(p), Binomial(m,p), Multinomial; Lecture and tutorial (5 hr)
Week 05 Estimating the minor allele frequency in a Hardy-Weinberg (HW) equilibrium, Comparing estimators: MSE and unbiased estimators, Standard error; Consistent estimator, Estimated MSE, Studying estimators: Poisson, HW problem estimators; Lecture and tutorial (5 hr)
Week 06 Comparing the two HW estimators using the delta-method; Asymptotic expression for the moment estimator from the HW problem; Parametric bootstrap, Conditional expectation; Lecture and tutorial (5 hr)
Week 07 Expected value and variance of the conditional expectation; Conditional variance and the variance of the conditional expectation, Properties of CDF, Continuity of probability measure, Continuous RVs; Continuous RVs, Uniform, exponential, Gamma and normal distributions, Quantiles; Lecture and tutorial (5 hr)
Week 08 Quantiles, Functions of a RV, Sampling given a uniform sample; Examples of functions of RVs, Joint distribution function/density, Marginal density; 2-d uniform distribution, Standard bivariate normal, Independent RVs, Bivariate normal are independent iff rho=0; Lecture and tutorial (5 hr)
Week 09 Conditional density, Sampling from 2-d distributions, Sum of jointly continuous RVs, Sum of independent Gamma RVs, Sum of independent exponential RVs; Beta distribution, Density of a quotient, Distribution of a function of a random point; Deriving the density of a sum from a 2-d transformation, General bi-variate normal, Rayleigh density, Box-Muller sampling, Quantile-Quantile Plot; Lecture and tutorial (5 hr)
Week 10 Quantile-Quantile and probability plots, Extrema and Order Statistics; Extrema and Order Statistics, Expectation of a continuous RV: Gamma, U(0,1) order statistics and its connection with probability plots, Cauchy, Normal, Function of a continuous RV; Lecture and tutorial (5 hr)
Week 11 Properties of expectation and variance, Variance of the Normal and Gamma distributions, Covariance and Correlation Coefficient, Correlation of bivariate normal; Markov, Chebyshev and the LLN revisited, Estimation in the continuous case, Estimating the normal parameters, Conditional expectation and variance; Conditional expectation and random sums in a more general context, Continuous analog of Law of Total Probability, Prediction; Lecture and tutorial (5 hr)
Week 12 Prediction, MGFs, moments and derivatives of the MGF; MGFs: of a sum of RVs, uniqueness, weak convergence, Continuity Theorem, CLT, Characteristic Functions; Confidence Intervals: motivation, definition and examples; Lecture and tutorial (5 hr)
Week 13 Another CI example and deriving CIs for a location parameter (with normal example); CIs: approximate based on parametric bootstrap, CIs for: a scale parameter, ratio of variances of normal distributions Lecture and tutorial (5 hr)
Weekly Computer lab Computer laboratory (1 hr)

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. construct appropriate statistical models involving random variables for a range of modelling scenarios. Compute (or approximate with a computer if necessary) numerical characteristics of random variables in these models such as probabilities, expectations and variances
• LO2. fit such models in outcome 1. to data (as appropriate) by estimating any unknown parameters
• LO3. compute appropriate (both theoretically and computationally derived) measures of uncertainty for any parameter estimates
• LO4. assess the goodness of fit (as appropriate) of a fitted model
• LO5. apply certain mathematical results (e.g. inequalities, limiting results) to problems relating to statistical estimation theory
• LO6. prove certain mathematical results (e.g. inequalities, limiting results) used in the course.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes Graduate qualities
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered

Additional information

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory
• Follow safety instructions in your manual and posted in laboratories
• In case of fire, follow instructions posted outside the laboratory door
• First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
• As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.