lecture05 - ECE 5510: Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 5 Today: (1) Expectation, (2) Families of discrete r.v.s HW 2 due on Thursday. Office hours today until 1:15 pm. Todays reading: 2.5, 2.7, 2.8. Thursdays reading: 3.1-3.5, 2.6, 3.7 0.1 Recap of Critical Material A random variable (r.v.) is a mapping, X : S R , where the range is S X R . For a discrete r.v. the range S X is countable. pmf: P X ( x ) = P [ { s S : X ( s ) = x } ] = P [ X ( s ) = x ] = P [ X = x ]. CDF, F X ( x ) is defined as F X ( x ) = P [ { X : X x } ]. 0.2 Expectation Section 2.5 of Y&G. Defn: Expected Value The expected value of discrete r.v. X is E X [ X ] = summationdisplay x S X xP X ( x ) E X [ X ] is also referred to as X . It is a parameter which describes the center of mass of the probability mass function. Note: Y&G uses E [ X ] to denote expected value. This is somewhat ambiguous, as we will see later. The first (subscript) X refers to the pmf well be using, the second X refers to what to put before the pmf in the summation. Example: Bernoulli r.v. Expectation What is the E X [ X ], a Bernoulli r.v.? E X [ X ] = summationdisplay x S X xP X ( x ) = summationdisplay x =0 , 1 xP X ( x ) = (0) P X (0)+(1) P X (1) = p ECE 5510 Fall 2009 2 Example: Geometric r.v. Expectation What is the E T [ T ], a Geometric r.v.?...
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lecture05 - ECE 5510: Random Processes Lecture Notes Fall...

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