## 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

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Additionally, you can easily link MyTimetable to a calendar app on your phone, and schedule changes will be automatically updated in your calendar. You can also choose to receive email notifications about schedule changes. You can enable notifications in Settings after logging in.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

**Note:** Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## 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. A partial grade can be transferred from the previous year

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

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.

Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

Enrolment for the fall opens in July

Enrolment for the spring opens in December

See this page for more information about deadlines and enrolling for courses and exams.

**Note:**

It is mandatory to enrol for all activities of a course that you are going to follow.

Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## Contact

e.w.van_zwet@lumc.nl

## Remarks

**Software**

Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.