Lect6_L6_L7_handout1

Lect6_L6_L7_handout1 - 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
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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 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 ---- rv, taking possible values ----
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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 ( ctd) What’s P ( X = 0 ) ? P ( X = 1 ) ?
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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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lecture 6, Discrete Random Variables I Random Variable Discrete Probability Distributions Expected Value Variance lecture 6, Discrete Random Variables I Discrete Probability Distributions Probability Distribution of X 0 1 2 3 4 0.0 0.1 0.2 0.3 0.4 Probability distribution of X X= # heads p(x)
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