infinitely independent tosses of a coin with PHeads p Sample space infinite

Infinitely independent tosses of a coin with pheads p

This preview shows page 35 - 48 out of 52 pages.

infinitely independent tosses of a coin with P(Heads) = p Sample space: infinite sequences of H and T Random variable X : number of tosses until the first Heads
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Probability & Statistics Discrete Random Variables Common Random Variables Bernoulli Uniform Binomial Geometric Expectation 12/23 Geometric Random Variable Geometric with parameter 0 < p 1 Experiment: infinitely independent tosses of a coin with P(Heads) = p Sample space: infinite sequences of H and T Random variable X : number of tosses until the first Heads Geometric random variable p X ( k ) = (1 - p ) k - 1 p
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation 13/23 5-min Break!
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 14/23 1 Discrete Random Variables 2 Common Random Variables 3 Expectation The Mean Expected Value Rule The Variance
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 15/23 The Mean Mean The expected value ( also called the expectation or the mean) of a random variable X , with PMF p X , is defined by E [ X ] , X x xp X ( x )
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 16/23 Bernoulli Random Variable Bernoulli random variable Bernoulli with parameter p [0 , 1] X = ( 1 w.p. p 0 w.p. 1 - p
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 17/23 Uniform Random Variable Consider a uniform random variable on [0 , 1 , . . . , n ]
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 18/23 Linearity of Expectations Linearity of Expectations Given a random variable X : E [ aX + b ] = a E [ X ] + b
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 19/23 Expected Value Rule Let X be a random variable, and Y = g ( X ) be a function of X . By definition: E [ g ( X )] = E [ Y ] = X y yp Y ( y )
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 19/23 Expected Value Rule Let X be a random variable, and Y = g ( X ) be a function of X . By definition: E [ g ( X )] = E [ Y ] = X y yp Y ( y ) Expected Value Rule E [ g ( X )] = X x g ( x ) p X ( x )
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 20/23 The Variance Average distance from the mean?
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 20/23 The Variance Average distance from the mean? Variance Variance is the expected value of the random variable g ( X ) = ( X - E [ X ]) 2 : var ( X ) = E [( X - E [ X ]) 2 ]
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Probability & Statistics Discrete Random Variables Common Random Variables Expectation The Mean Expected Value Rule The Variance 20/23 The Variance Average distance from the mean?
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  • Fall '14
  • LuisDavidGarcia-Puente
  • Probability theory, Nguyen Minh Huong

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