# PostLecture01EN.pdf - Probability Theory Adriana Gabor...

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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.

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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)
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 .

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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 .
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 .

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• Fall '19
• Probability theory, 3 minutes, 5 minutes, Mr. Z, 0 5 k

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