Probability and Statistics

Course description. This is an introduction to probability theory and basic statistics with an emphasis on problem solving. Topics include combinatorial probability, random variables and distributions, expectation and variance, joint distributions and independence, limit theorems, and fundamentals of statistical inference.

Attendance: Attendance at lectures and practices is not mandatory.

Midterm exam, retake midterm: During the semester there will be one midterm exam consisting of 6 problems, each worth 20 points. The duration is 90 minutes. Students receive the semester signature (i.e., they become eligible to register for the final exam) if they achieve at least 40 points on the midterm. There will be one retake opportunity during the teaching period, and an additional paid retake opportunity (second retake) in the make-up week before the final exam period. Both occasions can be used either to make up for a missed midterm or to improve the result of a failed one. The first retake (and only that) may also be used to improve an already successful result. If a student rewrites an earlier midterm at a retake, then the new score will be valid even if it is lower than the previous one. There is one exception: a failed improvement attempt cannot revoke the semester signature. Thus, if a student has already achieved at least 40 points on the original midterm but scores fewer than 40 points on an improvement attempt, their score will be adjusted to 40 (and the semester signature remains valid). If a student attends a retake session (and accepts the problem sheet), this counts as an official attempt at the exam, and the above rules apply. No prior registration in Neptun (or elsewhere) is required for the retake. Participation is entirely voluntary, regardless of whether the student intends the attempt as a make-up or as an improvement.

Second retake midterm: Students whose midterm result is still unsuccessful after the retake midterm may take it again at the second retake opportunity. This occasion appears in Neptun under the name Paid retake (formerly Signature-retake exam). Registration for this exam must be completed in Neptun and is subject to an additional administrative fee. If a student does not register in Neptun, we cannot record the obtained signature in the system. Therefore, students who fail to register in Neptun are not allowed to participate in this retake.

Signature obtained in a previous semester: Students who already hold a valid course signature from VISZAB04 in a previous semester, and who have registered again for the regular lecture and practice classes this semester (i.e., not only for the exam course), may attempt the midterm again in order to improve their previous midterm result. The following conditions apply: If the student again fulfills the requirements for obtaining the signature, then the new midterm score will count towards the final grade (whether it is better or worse than the earlier one). If the student does not fulfill the signature requirements again, the previously obtained signature remains valid, but for the purpose of the final grade only the minimum score required for the signature (40 points) will be taken into account. If a student with an existing signature attends at least one midterm in the current semester, this will be considered an attempt to fulfill the signature requirements again, and the above rules apply. Otherwise, the most recent semester in which the student attempted the requirements will be taken into account. Please note that there will be minor changes in the course material compared to last year, and the current material if the semester is also relevant for students enrolled only in the exam course.

Final exam At the end of the semester, students who hold a valid course signature must take a written exam in order to obtain the final grade. The exam consists of 6 problems, each worth 20 points, including one theoretical question. The duration is 100 minutes. If the exam score is below 40 points, the exam is failed and the final grade is fail, regardless of the midterm results. The grade is also changed to fail if a student who has already passed the exam attempts to improve their grade and obtains fewer than 40 points on the new attempt. In the case of a repeated exam, the midterm results remain valid. Only students with a valid course signature may register for the exam. Registration must be completed in Neptun. Please note that Neptun only allows us to record results for students who have registered, therefore we cannot administer the exam for students who fail to register. For a successful exam, the final score is calculated from the midterm and exam results according to the following formula:

Final score = 0.4*Midterm score+0.6*Exam score

The grade is assigned based on the final score as follows: [40;55): Satisfactory (pass), [55;70): Fair (average),[70;85): Good, [85,100]: Excellent. If the final score corresponds to at least Satisfactory, students may request an optional oral examination during the review session. The oral exam may change the grade by one step, either upwards or downwards. There is no requirement to wear formal attire at the exam for this course.

The available working time is 90 minutes for the midterm and 100 minutes for the final exam.

The rules for conducting the midterm and the exam are the same.

Both consist of 6 problems worth 20 points each; in the exam one problem is theoretical.

Work may only be done on pre-stapled sheets, including scratch paper.

Leaving the room is not allowed during the first 30 minutes; after that, entering the room is no longer permitted.

The upper right corner of the first page must contain: the student's name, Neptun code, and for the midterm the name of the exercise class instructor according to Neptun.

Numerical answers should be rounded to 4 significant digits.

To receive full credit, the complete solution process must be shown. The properties and theorems used in each step must also be indicated.

Permitted aids: a calculator (without graphical display), the student's own knowledge, and the distribution tables provided by the instructors. In the exam, if a problem requires it, additional formula sheets/tables distributed by the instructors (see end of the course notes supplement) may also be used.

Mobile phones and any other unauthorized electronic devices must be placed in the student's bag or handed over to the invigilators at the instructor's desk. Possession of such a device during the exam will be considered an attempt at cheating, regardless of whether it was used.

Lectures

Lecturer: Bence Csonka, Email: csonkab@edu.bme.hu

• Monday 12:15–13:45 — Room IE220
• Tuesday 10:15–11:45 — Room IE220

Practices

Exercise instructor: Humara Khan, Email: humara.khan@edu.bme.hu

• Monday 14:15–16:00 — Room IB140
• Wednesday 14:15–16:00 — Room E402
• Friday 12:15–14:00 — Room E406

• Sample midterm, Sample midterm solution
• Sample midterm 2 , Sample midterm 2 solution
• Midterm 2025, 2025 Midterm solution
• Retake Midterm 2025, 2025 Retake Midterm solution
• Joint continuous distribution example
• Common distribution table
• Standard Phi Distribution
• Probability theory part lecture notes: Probability Notes
• Statistics part lecture notes: Statistics Notes
• Recommended: Grinstead & Snell, Introduction to Probability
• Recommended: Ross, A First Course in Probability

• Midterm: October 27, 8:00—10:00
• Retake midterm: November 10, 8:00—10:00
• Second retaken midterm: December 15

Week	Dates	Lecture topic	Solutions of practices	Notes
1	Sep 8–14	Course intro; sample spaces; Conditional probability; Bayes’ rule; independence	Practice 1 Practice 1 solution	Lecture 1 Lecture 2
2	Sep 15–19	Combinatorics, Random variables	Practice 2 Practice 2 solution	Lecture 3
3	Sep 22–27	Discrete random variables, Poisson-, Geometric-, Binomial distribution, expected value	Practice 3 Practice 3 solution	Lecture 4 Lecture 5
4	Sep 29–Oct 3	Expected value, transformed disrcete random variables, deviation, continuous random variables	Practice 4 Practice 4 solution	Lecture 6 Lecture 7
5	Oct 6–10	Break	Practice 5 Practice 5 solution
6	Oct 13–17	Exponential and Uniform Distributions. Distribution Transformations and Expected Value in the Continuous Case	Practice 6 Practice 6 solution	Lecture 8 Lecture 9
7	Oct 20–24	Normal Distribution, de Moivre-Laplace Theorem, Central Limit Theorem, Random vectors	Practice 7 Practice 7 solution	Lecture 10 Lecture 11
8	Oct 27–October 31	Random vectors, joint continuous distribution, covariance, correlation	Practice 8 Practice 8 solution	Lecture 12 Lecture 13
9	Nov 4–Nov 10	Linear regression, conditional probability in continuous case, conditional expected value	Practice 9 Practice 9 solution	Lecture 14 Lecture 15
10	Nov 10–16	Examples for law of total expectation, sum of normal distributions	Practice 10 Practice 10 solution	Lecture 16
11	Nov 17–23	Probability inequalities, basic concepts of statistics		Lecture 17 Lecture 18
12	Nov 25–Dec 1
13	Dec 2–8
14	Dec 9–15