PostLecture01EN.pdf - Probability Theory Adriana Gabor...

  • No School
  • AA 1
  • 24

This preview shows page 1 - 6 out of 24 pages.

Probability Theory Adriana Gabor [email protected] Room H 11.27 Instructions for using the slides 1. One day before the lecture the handouts will be posted on blackboard. The handouts contain empty pages to write down the proofs and the examples that are discussed on the whiteboard during the lecture. 2. After the lecture a more detailed version of the handouts (called post lecture notes) will be posted on blackboard (containing sketches of the examples and proofs). 3. During lectures, we will discuss different examples than in the book. Therefore the slides are suplementary material and do not replace the book. Some examples on the slides are meant to be studied at home.
Image of page 1

Subscribe to view the full document.

Homework 1. Homework: Has to be handed in every Tuesday, before 11.00 in the folders in front of H11.04. It is allowed to work in groups of 2. The homework can be found in the folder Homework on blackboard. 2. Every homework contains 1-2 exercises from previous exams about the subject. These exercises do not have to be handed in, but they will give you an indication of the level that can be expected at the exam. Lecture 1: Chapters 2.3 (until Mixed Distributions)- Chapter 2.4 (beginning)
Image of page 2
Plan for lecture 1 I Rehearsal of basic concepts of probability theory I Continuous random variables: definition, cumulative distribution function, probability density, calculation of probabilities I Expected value of a continuous random variable. The sample space I We perform an experiment . I We call the results of the experiment outcomes . I An simple event is an event that consists of an outcome. An event that is not simple is called a compound event. I The sample space is the set of all simple events. We denote the sample space by S .
Image of page 3

Subscribe to view the full document.

Example Example 1 I Experiment: We independently toss a coin two times, having equal chances for heads and tails. I { HH } , { TH } , { HT } , { TT } are simple events. I The tossing of at least one tail is a compound event: { HT , TH , TT } I Sample space: S = { HH , TH , HT , TT } The probability measure P The probability measure P ( · ) assigns to every event A a number P ( A ) . A probability measure has the following properties: I P ( A ) 0, for every event A . I P ( S ) = 1 I For all events E 1 , E 2 ... with E i E j = : P ( i = 1 E i ) = X i = 1 P ( E i ) . Example 1- continued: The probability of tossing tail at least one time: P ( HT or TH or TT ) = P ( HT ) + P ( TH ) + P ( TT ) = 3 4 .
Image of page 4
Definition of a random variable Definition A random variable is a real function on the sample space: X : S 7→ R . Example 1- continued Random variable X : S 7→ R , X = the number of heads. { TT } X -→ 0 { HT } X -→ 1 { TH } X -→ 1 { HH } X -→ 2 P ( X = 1 ) = P ( { HT , TH } ) = 2 4 = 1 2 P ( X = 0 ) = P ( TT ) = 1 4 P ( X = 2 ) = P ( HH ) = 1 4 Random variable Y : S 7→ R , Y = ( the number of heads ) 2 .
Image of page 5

Subscribe to view the full document.

Image of page 6
  • Fall '19
  • Probability theory, 3 minutes, 5 minutes, Mr. Z, 0 5 k

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes