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Understand why probability theory and statistics play an
important role in reasoning in which randomness occurs.
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Know sample spaces, events, and the probability of an
event. Know simple rules to combine events, and the probabilities of
these events.
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Know the concept of the conditional probability of an
event given another event. Know the law of total probability, and how
the formula of Bayes can be derived from this. Be able to use the
formula of Bayes. Know the definition of two (and more) events being
independent.
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Know the definition of a discrete random variable, and
know a number of classical discrete distributions, such a the Bernoulli
distribution, the Binomial distribution and the Geometrical
distribution. Know the definition of the distribution function of a
random variable.
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Know the definition of a continuous random variable, and
know the relation between the probability density function and the
distribution function of a continuous random variable. Understand the
intrinsic difference between discrete and continuous random variables.
Know a number of classical continuous distributions, such a the
Exponential distribution, the Uniform distribution and the Normal
distribution
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Know the concept of a pseudo random generator. Know that
every function with the properties of a distribution function is a
distribution function of some random variable, and be able to use this
to perform simulations.
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Know the definition of the expectation and the definition
of variance of a random variable in case the random variable is discrete
and in case the random variable is continuous.
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Be able to derive easy but handy properties of the
expected value and of the variance of a random variable, such as E(X+Y)
= E(X)+E(Y), E(aX+b) = aE(X)+b, … .
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Know multidimensional distributed random variables (both
in the discrete and continuous case), their relation to the marginal
distributions. In the continuous case: know the concepts of probability
density function and distribution function of two (or more) random
variables, and their relation. Know the concept of independence of two
(or more) random variables, and (in the continuous case) its relation to
the distribution functions of the multidimensional random variable under
consideration and its marginal distributions.
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Know some elementary properties of the expectation of a
multidimensional random variable. Know and understand the concept of
covariance and the correlation coefficient of two random variables, and
the relation with independence. Know the concept of a sample from a
distribution. Know the convolution of n random variables.
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Know examples of the 1- and 2-dimensional Poisson
Process, and some related distributions such as the Poisson distribution
and the Pareto distribution.
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Know the concept of the distribution of a random
variable, given another random variable, and the concept of conditional
expectation. Be able to use these concepts in various situations. Know
the Raleigh distribution.
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Know the content of and be able to work with Cebyshev’s
inequality and the Law of Large Numbers; Be able to give a proof of the
Weak Law of Large Numbers, using Cebyshev’s inequality.
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Know the content of and be able to work with the Central
Limit Theorem; be able to approximate probabilities related to the mean
of any sample distribution using the Central Limit Theorem.
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Know graphical representations of the data such as the
histogram, kernel estimators, empircal distribution function and the
box-plot.
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Know numerical representations of data such as its mean,
variance, standard deviation, MAD and order statistics.
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Understand that an underlying model is needed in order to
perform a statistical analysis and that the histogram and kernel
estimators can be used to estimate probabilities of certain events. Be
able to model simple situations in which randomness occurs.
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Know the concept of the bootstrap, and be able to perform
a simple bootstrap experiment.
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Know the definition of an unbiased estimator of a
function of a parameter/parameters. Be able to determine whether a
given estimator is unbiased.
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Understand the role of the variance of an estimator when
comparing two estimators for the same function of a
parameter/parameters. Know and be able to work with the MSE (= mean
squared error) of an estimator.
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Know and be able to work with the concept of a maximum
likelihood estimator both in the case of a discrete and in the case of a
continuous random variable. Know the concepts of likelihood and
loglikelihood in both the discrete and in the continuous case.
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Know basic concepts of linear regression and least
squares estimation, and be able to apply this in an easy setting.
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Know and understand the basic concepts of testing of a
hypothesis, and knowing basic concepts such as nul-hypothesis,
alternative hypothesis, P-value, level of significance, critical region,
and the relations between these concepts.
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Know what errors of the first and second kind are, and
their relation to the above mentioned concepts.
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Know what the Student t-test is, know what the underlying
distribution of the testing variable is, its relation to the standard
normal distribution, and know when this test can be applied.
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Being able to apply the Student t-test in a simple linear
regression model.
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Know and be able to work with the concept of a confidence
interval.
- Know the relation between the
concepts of a confidence interval and the testing of a hypothesis.