Lect6_L6_L7_handout2

# Lect6_L6_L7_handout2 - lecture 6, Discrete Random Variables...

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lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Outline ± Random Variables ± Discrete Probability Distributions ± Expected Value and Variance of Discrete Random Variables The Binomial Distribution The Poisson Distribution

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lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Random Variable ± Review: An experiment is any process of observation with an uncertain outcome ± A random variable (rv in short) is used to describe some important aspect of the outcome. Eg, weather tomorrow (sunny, cloudy, rainy. ..) Eg: for a horse racing, a rv can be the ﬁnish time of the winning horse Eg: for a soccer game, a rv can be total goals, or time of the ﬁrst goal. .. ± Types of random variables: Categorical or Numerical Eg of categorical rv: weather tomorrow takes value in the categories: sunny, cloudy, rainy, etc. Eg of numerical rv: the ﬁnish time of the winning horse, or the total goals
lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Random Variable Numerical Random Variables ± Further classiﬁcation: Numerical rv ± Discrete rv Continuous rv . ± Discrete rv: Possible values can be counted or listed ± Continuous rv: May assume any numerical value in one or more intervals Eg of discrete rv: the total goals Eg of continuous rv: the ﬁnish time of the winning horse Focus on discrete rv in this chapter

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lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Random Variable Example: Coin tossing ± Toss a (fair) coin 4 times. ± Sample space = {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH,TTHT,TTTH,TTTT } ± All are equally like, hence each outcome has probability 1/16 ± X := # heads X is a ---- discrete rv, taking possible values ---- 0,1,2,3,4
lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Random Variable Example: Coin tossing ( ctd) What’s P ( X = 0 ) ? P ( X = 1 ) ? ± { X = 0 } consists of one outcome: TTTT, hence P ( X = 0 ) =1/16 ± { X = 1 } is an event consisting of 4 outcome: HTTT, THTT, TTHT, TTTH, hence P ( X = 1 ) =4/16, Similarly, P ( X = 2 ) = 6/16, P ( X = 3 ) = 4 / 16, and P ( X = 4 ) = 1 / 16 x 0 1 2 3 4 P ( X = x ) 1/16 4/16 6/16 4/16 1/16

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lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Discrete Probability Distributions Discrete Probability Distributions ± The probability distribution of a discrete rv is a table, graph or formula that gives the probability associated with each possible value that the variable can assume For each value x that X can assume, the probability P ( X = x ) is denoted by p ( x ) Eg: for the previous example, x 0 1 2 3 4 P ( X = x ) = p ( x ) 1/16 4/16 6/16 4/16 1/16
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## This note was uploaded on 12/30/2009 for the course ISOM ISOM111 taught by Professor Anthonychan during the Fall '09 term at HKUST.

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Lect6_L6_L7_handout2 - lecture 6, Discrete Random Variables...

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