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10_DiscreteRVs-2_Expectation_packed

# 10_DiscreteRVs-2_Expectation_packed - 2 Discrete Random...

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2. Discrete Random Variables Part II: Expecta±on ECE 302 Spring 2012 Purdue University, School of ECE Prof. Ilya Pollak

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Expected value of X : DeFni±on Ilya Pollak
Expected value of X : DeFni±on E [ X ] is also called the mean of X Ilya Pollak

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Example: mean of a Bernoulli random variable Ilya Pollak
(Note: E[ X ] is not necessarily one of the values that X can assume with a non-zero probability!) Example: mean of a discrete uniform random variable Ilya Pollak

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Expected value: Center of gravity of PMF c x • Imagine that a box with weight p X ( x ) is placed at each x. • Center of gravity c is the point at which the sum of the torques is zero: Ilya Pollak
Expected value and empirical mean Many independent Bernoulli trials with p =0.2 Bernoulli random variables X 1 , X 2 , … E[ X n ]=0.2 Empirical average from many independent experiments close to the expected value (law of large numbers) Ilya Pollak

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What E[ X ] is NOT It’s not necessarily the most likely value of X It’s not even always the case that P( X =E[ X ])>0 It’s not guaranteed to equal to the empirical average Ilya Pollak
Two ways to evaluate E[ g ( X )] This is true because g ( x ) p X ( x ) x = g ( x ) y p X ( x ) { x | g ( x ) = y } y = y p X ( x ) { x | g ( x ) = y } P ( g ( X ) = y ) = p Y ( y )     y = yp Y ( y ) = E [ Y ] y Ilya Pollak

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Cau:on± In general, E[ g ( X )]≠ g (E[ X ]) Example± average speed vs average :me. Suppose you need to drive 60 miles. A very bad storm will hit your area with probability 1/2. If the storm hits, you will drive 30 miles/hour during the en:re trip; If the storm does not hit, you will do 60 miles/hour during the en:re trip. What’s the expected value of your speed? (1/2) · 30 + (1/2) · 60 = 45 mph Is the expected value of your travel :me 60/45 = 1 hr 20 min?
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10_DiscreteRVs-2_Expectation_packed - 2 Discrete Random...

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