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• Introduction to Probability, fall 2013
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# Introduction to Probability with MATLAB, spring 2014

5 cr

### Type

This course is aimed for international students with a background in Computer Science or an application discipline, such as Bioinformatics Master´s program (MBI). No background in statistics or probabilistics is required.

The course can be included in advanced studies in the MBI program.

### Contents

The course starts from the elements of probability calculus, and will cover topics and concepts such as:

• probability, joint probability, and conditional probability
• random variable
• some elementary distributions such as binomial, uniform, exponential, and normal
• probability density function (pdf), cumulative distribution function (cdf)
• expected value, variance, and other useful statistics of a distribution

MATLAB is used as an educational tool in exercises, to obtain a practical understanding of random variables and probability distributions. (This is not, however, a course about MATLAB programming. The course will cover MATLAB only to such extent as is needed to use it for studying probability. Also, no previous familiarity with MATLAB is needed.)

### Prerequisites

Elementary calculus

### Lectures and exercises

The course lasts 6 weeks, starting 21.1.2014 and ending 26.2.2014.

Weekly schedule:

• Tue 12-14: lecture in room C128
• Tue 14-16: exercises in room C128
• Wed 12-14: lecture in room CK111
• Wed 14-16: exercises in room C128

The exercise sessions are instructed by Brittany Rose.

There are no deadlines for the exercises since no credits are given for them. Doing the exercises is for learning the topics. However, you may return your answers (on paper or in e-mail) if you wish to have comments on them.

### Lecture topics

• Lecture 2 (22.1.) (More combinatorics: binomial coefficient, sampling); non-equiprobable finite probability space
• Lecture 3 (28.1.): Product rule; conditional probability; Bernoulli trial (= repeated independent experiments with same probability of success each time)
• Lecture 4 (29.1.): More about conditional probability; law of total probability; Bayes's rule (inverting the conditional probability)
• Lecture 5 (4.2.): Random variable; discrete distribution; binomial and geometric distribution; expected value and median
• Lecture 6 (5.2.): Discrete distributions and expected value continued
• Lecture 7 (11.2.): Continuous distributions; uniform distribution; density function; empirical histogram
• Lecture 8 (12.2.): Continuous distributions continued. Pdf, cdf, expected value. Expected gain from a bet, linked to subjective probability.
• Lecture 9 (18.2.): Exponential distribution. Distribution of a sum. Multinomial coefficient and multinomial distribution.
• Lecture 10 (19.2.): Expected value of a function g(X). Variance and standard deviation.
• Lecture 11 (DRAFT) (25.2.): Markov's and Chebyshev's inequalities, and law of large numbers (G&S: Chapter 8).
• Lecture 12 (26.2.): Normal distribution and the central limit theorem. Distribution of a maximum (of any distribution).

Lecture slides are provided here if they exist. For blackboard lectures there may not be slides.

• There is a nice textbook by Grinstead and Snell available at
http://www.math.dartmouth.edu/~prob/prob/prob.pdf
This intro course will not cover everything in the book, but it provides good reading for deeper understanding.
• Important topics include the following:
• Chapter 1: Idea of random variable. Section 1.2: definitions and theorems.
• Chapter 2: Continuous randon variable. Density function and cumulative distribution function.
• Chapter 3: "A Counting Technique" = rule of product. Permutations, factorials, combinations, binomial coefficient. Bernoulli trial and binomial distribution. (Card shuffling is out of scope.)
• Chapter 4: Conditional probabilitiy. Definitions and theorems. Bayes' formula. (Joint density, beta density are out of scope.)
• Chapter 5: Discrete uniform, binomial, geometric, hypergeometric distribution; continuous uniform, exponential, normal distribution. (Other distributions are out of scope.)
• Chapter 6: Definition, interpretation and theorems of expected value and variance.
• (Chapter 7 is out of scope.)
• Chapter 8: Chebysev, and law of large numbers.
• Chapter 9: Central limit theorem (what it means and how it is used). Proof of CLT is out of scope.
• (Chapters 10—12 are out of scope.)
• "Out of scope" means it did not fit this one-period introductory course (and will not be required in the exam), but for further studies in probability, many of these chapters contain much useful reading.

### Exams

The course is passed by taking a single course exam, which is held at the general examination, Tuesday 4.3.2014 starting at 12.

General examinations are arranged at Exactum A111 and B123. The detailed information about the place can be found on the exam day from the note on the door of the auditorium.

Note: Although the general examination (where many different exams can be taken) lasts four hours, this course exam lasts 2 hours (you have to return your answers in two hours).

You can bring the following tools to the exam: a calculator; an A4 sheet of handwritten notes (you can use both sides of the paper); the MAOL book of tables. At the exam we will provide you this printed collection of tables.

Update 13.3.2014: The exam has been graded. The results will be registered to Weboodi soon, and will appear here. For any questions about the grading, contact the lecturer.

### Registration

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