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# Lecture8 - lecture 8 Discrete Random Variables I lecture 8...

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lecture 8, Discrete Random Variables I 1 / 24 Random Variable Discrete Probability Distributions Expected Value Linearity of Expectations Variance lecture 8, Discrete Random Variables I Outline 1 Random Variable 2 Discrete Probability Distributions 3 Expected Value Linearity of Expectations 4 Variance

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lecture 8, Discrete Random Variables I 2 / 24 Random Variable Discrete Probability Distributions Expected Value Linearity of Expectations Variance lecture 8, Discrete Random Variables I Random Variable Random Variable A random variable is a function, say X , from a sample space S to the real numbers R : Classification of random variable: A random variable is said to be discrete if it takes on finitely or countably many values E.g.: number of heads when tossing a coin 4 times continuous if it may assume any value in one or more intervals E.g.: finish time of the winning horse in a horse racing Focus on discrete random variable in this chapter Convention: use capital letters for random variables, and lowercase letters for the possible values
lecture 8, Discrete Random Variables I 3 / 24 Random Variable Discrete Probability Distributions Expected Value Linearity of Expectations Variance lecture 8, 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 E.g., X ( HHHH ) = 4 , X ( HHHT ) = 3 , etc. The possible values that X can take are , hence X is a discrete random variable

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lecture 8, Discrete Random Variables I 4 / 24 Random Variable Discrete Probability Distributions Expected Value Linearity of Expectations Variance lecture 8, Discrete Random Variables I Discrete Probability Distributions Discrete Probability Distributions Let X be a discrete random variable. The probability distribution of X identifies the probability associated with each possible value that X can take. The probability distribution can be represented by a table, or a graph, or a formula. For each value x that X can take, the probability P ( X = x ) is denoted by p ( x )
lecture 8, Discrete Random Variables I 5 / 24 Random Variable Discrete Probability Distributions Expected Value Linearity of Expectations Variance lecture 8, Discrete Random Variables I Discrete Probability Distributions Example: Coin tossing, ctd X := # heads when tossing a (fair) coin 4 times. What’s the probability distribution of X ? The possible values X can take are { 0 , 1 , . . . , 4 } Need to find for each x = 0 , 1 . . . , 4, the probability P ( X = x ) { X = 0 } : consists of one outcome: TTTT, hence P ( X = 0 ) =1/16 { X = 1 } : consists 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 Hence the probability distribution of X can be represented by the following table x 0 1 2 3 4 P ( X = x ) 1/16 4/16 6/16 4/16 1/16

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lecture 8, Discrete Random Variables I 6 / 24 Random Variable Discrete Probability Distributions Expected Value Linearity of
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