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## Probability Theory

2019-2020
Dit vak wordt in het Engels aangeboden. Omschrijvingen kunnen daardoor mogelijk alleen in het Engels worden weergegeven.

### Doel vak

After this course:

1. Students know the three axioms of probability and associated
basic propositions and are able to use these to compute probabilities of
events.
They understand the concept of independence and can compute conditional
probabilities (directly or with Bayes' rule).
2. Students can compute probability mass functions and density
functions, cumulative distributions, expectations and variances of
discrete and continuous random variables. They also know important
discrete and continous probability distributions (Bernoulli, binomial,
Poisson, geometric, negative bionomial, hyper geometric, uniform, normal
and exponential distribution) and their properties and know in which
context these distributions occur.
3. Students 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. Students can compute the probability distribution of a function of
a random variable or a random vector.
5. Students know and can apply the central limit theorem.

### Inhoud vak

We study experiments in which randomness plays a role. We first consider
discrete probability experiments: 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 reasonable probabilities to events, such that
the three axioms of probability are satisfied. We compute probabilities
in these situations and consider associated concepts such as
independence,
conditional probabilities, random variables and important discrete
probability distributions e.g. 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 consider 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 deal with
independence, conditional distributions and expectations, distributions
and expectations of functions of random vectors and covariance. We study
the Central limit theorem and normal approximations to Binomial
distributions.

### Onderwijsvorm

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

### Toetsvorm

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

### Literatuur

Anderson, D.F, Seppäläinen, T, and Valkó, B, Introduction to
Probability, Cambridge University Press, 2018.

### Doelgroep

Mathematics and BA, year 1 and premaster Mathematics

### Aanbevolen voorkennis

-First year calculus courses for BA or Mathematics. Mathematics students
have their second Calculus course simultaneously with Probability
Theory.
-Sets and Combinatorics (BA) or Basic Concepts in Mathematics
(Mathematics). If you were unable to take the course Basic Concepts
in Mathematics or Sets and Combinatorics, e.g. because you are a part
time
student, it is recommended that you study Appendix B and C of the book
before the
start of the course.

### Algemene informatie

Vakcode X_400622 6 EC P4+5 200 Engels Faculteit der Bètawetenschappen dr. C.M. Quant dr. C.M. Quant dr. C.M. Quant

### Praktische informatie

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