Lecture_7.pdf - Lecture 7 Jonathan UC San Novak Diego Math 183 Oct 14 2019 Example You are playing On each toss lose a if the model the game D=

Lecture_7.pdf - Lecture 7 Jonathan UC San Novak Diego Math...

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Lecture 7 Jonathan Novak UC San Diego Math 183 Oct 14 , 2019
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Example : You are playing a game in which a fair coin is flipped 3 times . On each toss , you win a dollar if the coin lands heads , and lose a dollar if the coin lands tails Analyze this game Solution : We start by building a probability space ( d. F , P ) to model the game We take D= { AHH , HHT , HTH , TH H , HTT , THT , TTH , TTT } , which represents all possible outcomes , F - - { Eel } , which represents all events , and IP ( E ) = FIT because the coin is fair
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Now let X :D IR be the random variable defined by X ( w ) = winnings if w occurs . Then XCHHH ) =3 X( HTT ) = - l X ( HHT ) - - I X( THT ) - - - I XCHTH ) - - I XLTTH ) - - - I XITHH ) - - I XITTT ) =-3 , so Supp ( X ) - E - 3 , - I , 1,33
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The probability mass function of X is thus I , if k =-3 Zz , if k = - I k " " " ÷÷÷÷ . . . . . The CDF of X is TIKI =P ( X ' x ) , so F- Cx ) - 2. p . ( x ) . Kc - Supp ( x ) n f - o , x ]
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Since X has finite support , its graph is a step function : ¥ : - T.is It remains to write down a formula for this function . .
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Last Time : Random Vectors Definition : Given a probability space ( D , I , P ) and a positive integer k , a K - dimensional random vector is a function X. r - spi . Definition : The support of X is the set of its outputs : Supp ( X ) = Exe Rk : Xcws - - x for some wed ) . X is said to be discrete if Supp ( X ) is finite or countably infinite , and continuous if Supp ( X ) is uncountable
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- Let X - ( X .
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