## Admission Requirements

None.

## Description

The first half of this course is an introduction to probability theory. We start with the definition of a probability distribution, and discuss conditional probabilities and (conditional) independence. Next, we introduce a number of well known and much used probability distributions and consider joint distributions of multiple, dependent variables. We define the expectation, variance and covariance and finally we discuss two important Theorems: the Law of Large Numbers and the Central Limit Theorem.

The second half of the course is an introduction to statistics. On the basis of data we try to draw conclusions about the (random) process that generated those data. We discuss estimation theory where we attempt to find the probability distribution that “"best" fits the data. In particular, we consider the method of maximum likelihood. Next, we look at testing hypotheses, where we try to determine if the data are consistent with some hypothesis, or not.

## Course objectives

**Learning aims for the probability part**

Upon completing the course, the student is able to ...

understand the concepts of experiment, sample space, event and apply counting methods to compute probabilities of events in realistic examples.

calculate marginal, conditional and joint probabilities from contingency tables.

understand the concept of a random variable, distinguish discrete from continuous random variables and identify in realistic situations popular distributions such as the Bernoulli, Binomial, Geometric, Poisson, Normal and Exponential.

compute probabilities, the expected value and variance using known discrete distributions.

compute and interpret the expected value, variance, covariance and correlation of linear combinations of random variables.

compute probabilities and determine the percentiles of a normal distribution using the standard normal tables.

derive the marginal probability distribution, conditional distribution and the corresponding expected value and variance from the joint probability distribution of discrete random variables.

motivate the use of limiting theorems to compute probabilities in practical examples.

**Learning aims for the statistics part**

Upon completing the course, the student is able ...

to explain and assess the quality of estimators (bias, variance, mean squared error)

to use simulation and the bootstrap to evaluate statistical procedures

to estimate a statistical parameter by means of the method of moments and maximum likelihood

to apply large sample theory to assess the uncertainty of the maximum likelihood estimator

to explain the concepts of the null and alternative hypothesis, type I and type II errors, and power in the context of hypothesis testing

to calculate the type I error, type II error and power of a particular hypothesis test

to explain, calculate and interpret the p-value

to derive the score, likelihood and Wald tests and compute their associated confidence intervals

to combine the above mentioned concepts and methods to answer a research question

## Timetable

You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.

MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).

For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.

## Mode of Instruction

Lectures, problem sessions, self study.

## Assessment method

There are two written partial exams. The final grade in the course is the average of the two partial exams. However, a minimal grade of 5.5 (on a scale from 1 to 10) is required for both partial exams. If you score below 5.5 on either partial exam, the final grade cannot exceed 5. The grades for the partial exams will be rounded to a tenth of a point, while the final grade is rounded to half a point (except 5.5 which will be changed to 6). The student only needs to re-do the part(s) of the exam that they have not passed yet.

The homework does not contribute to the final grade.

## Reading list

Recommended:

Mathematical Statistics and Data Analysis. John A. Rice, Duxbury press (3-rd ed. 2007)

A Modern Introduction to Probability and Statistics F.M. Dekking; C. Kraaikamp; H.P. Lopuhaä; L.E. Meester (2-nd ed. 2007)

## Registration

It is the responsibility of every student to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.

Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.

Extensive FAQ's on MyStudymap can be found here.

## Contact

e.w.van_zwet@lumc.nl