Assignment 1
1 General information
The exercises of this assignment are meant to test whether or not you have sufficient knowledge to participate in the course. The first question checks that you remember basic terms of probability calculus. The second exercise checks your basic computer skills and guides you to learn some R functions. In the last three ones you will first write the math for solving the problems (you can, for example, write the equations in markdown or include a photo of hand written answers), and then implement the final equations in R (and then you can use markmyassignment to check your results). The last question checks that you have found the course book.
The maximum amount of points from this assignment is 3.
We prepared a quarto template specific to this assignment to help you get started. You can inspect this and future templates
- as a qmd file,
- as a rendered html file
- or as a rendered pdf file
or you can download all template qmd files and some additional files at templates.zip (also available on Aalto JupyterHub under /coursedata).
Reading instructions:
Grading instructions:
The grading will be done in peergrade. All grading questions and evaluations for this assignment are contained within this document in the collapsible Rubric blocks.
- The recommended tool in this course is R (with the IDE RStudio).
- Instead of installing R and RStudio on you own computer, see how to use R and RStudio remotely.
- If you want to install R and RStudio locally, download R and RStudio.
- There are tons of tutorials, videos and introductions to R and RStudio online. You can find some initial hints from RStudio Education pages.
- When working with R, we recommend writing the report using
quartoand the provided template. The template includes the formatting instructions and how to include code and figures. - Instead of
quarto, you can use other software to make the PDF report, but the the same instructions for formatting should be used. - Report all results in a single, anonymous *.pdf -file and submit it in peergrade.io.
- The course has its own R package
aaltobdawith data and functionality to simplify coding. The package is pre-installed in JupyterHub. To install the package on your own system, run the following code (upgrade="never" skips question about updating other packages):
install.packages("aaltobda", repos = c("https://avehtari.github.io/BDA_course_Aalto/", getOption("repos")))- Many of the exercises can be checked automatically using the R package
markmyassignment(pre-installed in JupyterHub). Information on how to install and use the package can be found in themarkmyassignmentdocumentation. There is no need to includemarkmyassignmentresults in the report. - Recommended additional self study exercises for each chapter in BDA3 are listed in the course web page. These will help to gain deeper understanding of the topic.
- Common questions and answers regarding installation and technical problems can be found in Frequently Asked Questions (FAQ).
- Deadlines for all assignments can be found on the course web page and in Peergrade. You can set email alerts for the deadlines in Peergrade settings.
- You are allowed to discuss assignments with your friends, but it is not allowed to copy solutions directly from other students or from internet.
- You can copy, e.g., plotting code from the course demos, but really try to solve the actual assignment problems with your own code and explanations.
- Do not share your answers publicly.
- Do not copy answers from the internet or from previous years. We compare the answers to the answers from previous years and to the answers from other students this year.
- Use of AI is allowed on the course, but the most of the work needs to by the student, and you need to report whether you used AI and in which way you used them (See points 5 and 6 in Aalto guidelines for use of AI in teaching).
- All suspected plagiarism will be reported and investigated. See more about the Aalto University Code of Academic Integrity and Handling Violations Thereof.
- Do not submit empty PDFs, almost empty PDFs, copy of the questions, nonsense generated by yourself or AI, as these are just harming the other students as they can’t do peergrading for the empty or nonsense submissions. Violations of this rule will be reported and investigated in the same way was plagiarism.
- If you have any suggestions or improvements to the course material, please post in the course chat feedback channel, create an issue, or submit a pull request to the public repository!
2 Basic probability theory notation and terms
This can be trivial or you may need to refresh your memory on these concepts (see, e.g. Aalto course First Course in Probability and Statistics). Explain each of the following terms with one sentence:
- probability
- probability mass (function)
- probability density (function)
- probability distribution
- discrete probability distribution
- continuous probability distribution
- cumulative distribution function (cdf)
- likelihood
3 Basic computer skills
This task deals with elementary plotting and computing skills needed during the rest of the course. You can use either R or Python, although R is the recommended language in this course and we will only guarantee support in R. For documentation in R, just type ?{function name here}.
- Please try to include as much code and output as needed, but as little as possible.
- Please make sure that the plots are properly labeled and are easily legible and understandable. This means
- they should have x- and y-labels,
- the text within should be of a size comparable to the size of the surrounding text and
- each plot should have a concise but descriptive caption or title.
- Please make sure to report a sensible number of digits when reporting numbers. You will get more precise instructions later on, but for now think independently about how many digits of your results are important for the assignment.
4 Bayes’ theorem 1
A group of researchers has designed a new inexpensive and painless test for detecting lung cancer. The test is intended to be an initial screening test for the population in general. A positive result (presence of lung cancer) from the test would be followed up immediately with medication, surgery or more extensive and expensive test. The researchers know from their studies the following facts:
- Test gives a positive result in \(98\%\) of the time when the test subject has lung cancer.
- Test gives a negative result in \(96\%\) of the time when the test subject does not have lung cancer.
- In general population approximately one person in 1000 has lung cancer.
Relatively high false negative (cancer doesn’t get detected) or high false positive (unnecessarily administer medication) rates are typically bad and undesirable in tests.
Here are some probability values that can help you figure out if you copied the right conditional probabilities from the question.
- P(Test gives positive | Subject does not have lung cancer) = \(4\%\)
- P(Test gives positive and Subject has lung cancer) = \(0.098\%\) this is also referred to as the joint probability of test being positive and the subject having lung cancer.
5 Bayes’ theorem 2
We have three boxes, A, B, and C. There are
- 2 red balls and 5 white balls in the box A,
- 4 red balls and 1 white ball in the box B, and
- 1 red ball and 3 white balls in the box C.
Consider a random experiment in which one of the boxes is randomly selected and from that box, one ball is randomly picked up. After observing the color of the ball it is replaced in the box it came from. Suppose also that on average box A is selected 40% of the time and box B \(10\%\) of the time (i.e. \(P(A) = 0.4\)).
6 Bayes’ theorem 3
Assume that on average fraternal twins (two fertilized eggs and then could be of different sex) occur once in 150 births and identical twins (single egg divides into two separate embryos, so both have the same sex) once in 400 births (Note! This is not the true value, see Exercise 1.6, page 28, in BDA3). American male singer-actor Elvis Presley (1935 – 1977) had a twin brother who died in birth. Assume that an equal number of boys and girls are born on average.