05e2a

05e2a - EE 464 Final Solutions Mitra, Spring 2005 TOTAL =...

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EE 464 Mitra, Spring 2005 Final Solutions TOTAL = 93 MEAN = 46.22 MAX = 74 MIN= 17 1. Problem 1 Basics (a) Give a defnition oF a probability space. Give an example oF a fnite-dimensional probability space and oF an infnite-dimensional probability space. A probability space is defned by the triple: , F , P ) where Ω is the sample space, F is the associated feld and P is the probability measure. An example oF a fnite dimensional probability space is the Fair coin example with: Ω = { H,T } , F = { φ, { H,T } , { H } , { T }} = the power set and the probability distribution given by P [ H ] = P [ T ] = 1 2 . An example oF an infnite dimensional probability space is a uniForm type distribution, that is, Ω = [0 , 1] , F = B [0 , 1] = the Borel sets For [0 , 1] , and For any sub-interval oF [0 , 1] , P [ a,b ] = b a . (b) Give a defnition oF a random variable. A random variable is defned on a valid probability space. It is a mapping X : Ω R (a mapping From the sample space to the real line) such that the set { X x } is a valid event ( i.e. an element oF F ) and P [ X = ] = P [ X = −∞ ] = 0 . 2. Problem 2 Characteristic ±unctions (a) Show that iF Φ( ω ) is a valid characteristic Function then | Φ( ω ) | 2 is also a valid characteristic Function. Can you describe how such a characteristic Function might come about? | Φ( ω ) | 2 = Φ( ω )Φ( ω ) = E [ e jωX ] E [ e jωX ] = E [ e jωX ] E [ e jωX ] Thus | Φ( ω ) | 2 is the characteristic Function oF X 1 X 2 where X 1 and X 2 are independent and identically distributed random variables. Clearly iF X 1 and X 2 are valid random variables, so is the sum and thus | Φ( ω ) | 2 is a valid characteristic Function. (b) Let f ( x ) and g ( x ) be two valid probability density Functions. ±orm the mixture probability density Function h ( x ) = ǫf ( x ) + (1 ǫ ) g ( x ) where 0 < ǫ < 1 . Determine the characteristic Function oF a random variable X with pdF h ( x ) . Φ( ω ) = i −∞ h ( x ) e jωx dx = i −∞ { ǫf ( x ) + (1 ǫ ) g ( x ) } e jωx dx = ǫ i −∞ f ( x ) e jωx dx + (1 ǫ ) i −∞ g ( x ) e jωx dx = ǫ Φ f ( ω ) + (1 ǫ g ( ω ) where Φ f ( ω ) is the characteristic Function oF a random variable with pdF f ( x ) and Φ g ( ω ) is the characteristic Function oF a random variable with pdF g ( x ) .
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05e2a - EE 464 Final Solutions Mitra, Spring 2005 TOTAL =...

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