Stat400Lec5(Ch2.1_2.3) - Stat 400 Lecture 5 Spring 2012...

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Stat 400 Lecture 5 Spring 2012 Review(Chapter 1) Experiment, events, set theoretic terminology Probability rules, Bayes’s theorem Independence Method of enumeration Today’s lecture(2.1, 2.2, 2.3) Random variable, space. Discrete random variable, probability mass function and its properties. Uniform distribution, Hypergeometric distribution Mathematical expectation (mean,variance,standard de- viation, moment) Properties of mathematical expectation (mean,variance) 1 of 13
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Stat 400 Lecture 5 Spring 2012 Random variable Definition: Given a random experiment with an outcome space S , a function X that assigns to each element s in S one and only one real number X ( s ) = x is called a random variable . The random variable X is discrete if the set of real values it can take if finite or countable, e.g. { 0 , 1 , 2 , 3 , ··· , } . Example: Paul rolls a defect dice with numbers 1,2,3,3,3,3 on six faces. Random experiment: which number? Random variable: X gives numbers to the possible out- comes. Definition: The space (support) of X is the set of real numbers { x : X ( s ) = x,s S } . In the book, the space of X is denoted by S , same as sample space. Space of X = S = In many instances, one can think of the space of X as being the outcome space, but ··· 2 of 13
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Stat 400 Lecture 5 Spring 2012 Example: Wheel of fortune. One spins a wheel and look at the angle of the pointer. If the angle is within [0 o , 180 o ) , then one gets 1000 dollars, otherwise he/she gets 0 dollars. Let X be the dollar amount the person gets. What is the outcome space ? What is the space for X ? Is X a discrete random variable? Probability mass function The probability mass function (p.m.f.) f ( x ) , for a dis- crete random variable X , is given by f ( x ) = P ( X = x ) , x S Example: Paul rolls a defect dice with numbers 1,2,3,3,3,3 on six faces. Which number? x
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This note was uploaded on 03/14/2012 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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Stat400Lec5(Ch2.1_2.3) - Stat 400 Lecture 5 Spring 2012...

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