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## Probability Theory

2018-2019

### Course Objective

After this course:

1. The student knows the three axioms of probability and associated
basic propositions and can use these to compute probabilities of events.
He understands the concept of independence and can compute conditional
probabilities (directly or with Bayes' rule).
2. The student can compute probability mass functions and density
functions, cumulative distributions, expectations and variances of
discrete and continuous random variables. He also knows important
discrete and continous probability distributions (Bernoulli, binomial,
Poisson, geometric, negative bionomial, hyper geometric, uniform, normal
and exponential distribution) and their properties and knows in which
context these distributions occur.distributions occur.
3. The student can work with jointly distributed random variables. He
can compute probabilities, use the concept of independence in this
context, compute marginal distributions, conditional distributions,
(conditional) expectations, covariances and correlations.
4. The student can compute the probability distribution of a function of
a random variable or a random vector.
5. The student knows and can apply the central limit theorem.

### Course Content

We study experiments in which randomness plays a role. We first consider
discrete probability experiments, that is experiments with a countable
number of possible outcomes. You can think of tossing dice, shuffling a
deck of cards, flipping coins etc. The possible outcomes form a set,
the so called sample space. Every subset of this sample space is an
event. We assign probabilities to events in a reasonable way, such that
the three axioms of probability are satisfied. We compute probabilities
in these situations and consider associated concepts like independence,
conditional probabilities, random variables and important discrete
probability distributions like the Bernoulli, Binomial, geometric,
hypergeometric, negative Binomial and Poisson distribution.

We then consider experiments with an uncountable number of possible
outcomes and continuous random variables. We treat a number of
well-known continuous distributions: the uniform, exponential, normal
and exponential distributions. We study joint distributions of several
(discrete or continuous) random variables. In this context we treat
independence, conditional distributions and expectations, distributions
and expectations of functions of random vectors and covariance. We study
the Central limit theorem and the normal approximation to the Binomial
distribution.

### Teaching Methods

lectures (4 hours per week) and tutorials (2 hours per week). There will
be biweekly short test during the tutorials.

### Method of Assessment

Midterm (40%) and final exam (40%), biweekly quizzes (20%), see Canvas
for the details.

### Literature

Sheldon Ross, A first course in probability, 9th edition.

### Target Audience

Mathematics and BA, year 1 and premaster Mathematics

### Recommended background knowledge

Calculus, Sets and Combinatorics (BA) or Basic Concepts in Mathematics
(Mathematics). If you were not able to follow the course Basic Concepts
in Mathematics or Sets and Combinatorics, e.g. since you are a part time
student, it is recommended to study the first chapter of Ross before the
start of the course.

### General Information

Course Code X_400622 6 EC P4+5 200 English Faculty of Science dr. C.M. Quant dr. C.M. Quant dr. C.M. Quant

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods Seminar, Lecture
Target audiences

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