Vakcode: wi2013mt
Vaknaam: Kansrekening en statistiek
ZIE OOK BLACKBOARD

Het betreft een College
ECTS studiepunten: 3

Tel.: 015-27 89215

Cursusjaar:
Semester: 
Coll.uren p/w: 
Andere uren: 
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(zie jaarindeling)

• Axiomatische opbouw. Klassieke kansdefinitie van Laplace, symmetrische kans ruimten. Eenvoudige combinatoriek. Voorwaardelijke kans, theorema van Bayes.
• Stochastische onafhankelijkheid, stochastische varia- belen, kansfunctie, kansdichtheid en verdelingsfunctie.
• Kansverdelingen: alternatief, binomiaal, Pascal, Poisson, uniform, exponentieel, normaal.
• Kansverdelingen vervolg: gezamenlijke verdeling van twee stochastische variabelen, onafhankelijkheid en voorwaardelijke verdelingen.
• Verwachting, variantie, covariantie, correlatie, momenten, ongelijkheid van Chebychev en centrale limietstelling. Populatie, steekproef en steekproefverdelingen.
• Schattingstheorie: zuiverheid, momentenschatters, meest aannemelijke schatters. Betrouwbaarheidsintervallen: definitie en eenvoudige voorbeelden.
• Toetsingstheorie: onbetrouwbaarheid, kritiek gebied, fout van de eerste en tweede soort, onderscheidingsvermogen.

• Larson. Introduction to probability theory and statistical inference, Wiley.
• Feller. An introduction to probability theory I, Wiley.
• Hogg en Craig. Introduction to mathematical statistics, Collier, McMillan.

• Understand why probability theory and statistics play an important role in reasoning in which randomness occurs.
• Know sample spaces, events, and the probability of an event. Know simple rules to combine events, and the probabilities of these events.
• 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.
• 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.
• 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
• 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.
• 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.
• 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, … .
• 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.
• 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.
• Know examples of the 1- and 2-dimensional Poisson Process, and some related distributions such as the Poisson distribution and the Pareto distribution.
• 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.
• 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.
• 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.
• Know graphical representations of the data such as the histogram, kernel estimators, empircal distribution function and the box-plot.
• Know numerical representations of data such as its mean, variance, standard deviation, MAD and order statistics.
• 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.
• Know the concept of the bootstrap, and be able to perform a simple bootstrap experiment.
• Know the definition of an unbiased estimator of a function of a parameter/parameters. Be able to determine whether a given estimator is unbiased.
• 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.
• 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.
• Know basic concepts of linear regression and least squares estimation, and be able to apply this in an easy setting.
• 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.
• Know what errors of the first and second kind are, and their relation to the above mentioned concepts.
• 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.
• Being able to apply the Student t-test in a simple linear regression model.
• 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.