FND-STA-Probability distribution

Probability Distribution

 

Keywords:  Nature of a probability distribution, important distributions, comparing observed and simulated probability distributions, Kullback-Leibler diveregence, the Kolmogorov-Smirnov test.

 

 

 

Abstract

Probability distributions are at the core of any statistical analysis, in which modelled distributions are compared with sampled distributions to relate an observation to our theoretical understanding. This unit introduces the principles, discusses Poisson, uniform, and normal distributions, and presents methods to compare distributions with each other and quantify the difference.

 

This unit ...

Prerequisites

You need the following preparation before beginning this unit. If you are not familiar with this material from courses you took previously, you need to prepare yourself from other information sources:

• Calculus: functions and equations; polynomial functions, logarithms, trigonometric functions; integrals and derivatives; theorem and proof.

You need to complete the following units before beginning this one:

• FND-STA-Probability (Probability)

 

Objectives

This unit will ...

• ... introduce basic concepts of probability distributions;
• ... demonstrate the Poisson, the uniform, and the normal distribution;
• ... teach how to visually and quantitatively compare them.

 

Outcomes

After working through this unit you ...

• ... can interpret observed events in terms of probability distributions;
• ... are familar with the Poisson, the uniform, and the normal distribution;
• ... can compare observed distributions against the normal distribution with qqnorm().
• ... can compare observed distributions against each other with qqplot().
• ... can use Kullback-Leibler divergence for discrete distributions, and ks.test() for continuous distributions to quantify differences.

 

Deliverables

• Time management: Before you begin, estimate how long it will take you to complete this unit. Then, record in your course journal: the number of hours you estimated, the number of hours you worked on the unit, and the amount of time that passed between start and completion of this unit.
• Journal: Document your progress in your Course Journal. Some tasks may ask you to include specific items in your journal. Don't overlook these.
• Insights: If you find something particularly noteworthy about this unit, make a note in your insights! page.

 

Contents

 

Task:

 

• Open RStudio and load the ABC-units R project. If you have loaded it before, choose File → Recent projects → ABC-Units. If you have not loaded it before, follow the instructions in the RPR-Introduction unit.
• Choose Tools → Version Control → Pull Branches to fetch the most recent version of the project from its GitHub repository with all changes and bug fixes included.
• Type init() if requested.
• Open the file FND-STA-Probability_distribution.R and follow the instructions.

 

Note: take care that you understand all of the code in the script. Evaluation in this course is cumulative and you may be asked to explain any part of code.

 

 

 

Further reading, links and resources

 

Notes

 

Self-evaluation

 

 

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If in doubt, ask! If anything about this learning unit is not clear to you, do not proceed blindly but ask for clarification. Post your question on the course mailing list: others are likely to have similar problems. Or send an email to your instructor.

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About ...
 
Author:

Boris Steipe <boris.steipe@utoronto.ca>

Created:

2017-08-05

Modified:

2017-08-05

Version:

1.0

Version history:

• 1.0 New material

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