lec7 - Random Variables Intuitive idea: If the value of a...

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Random Variables Intuitive idea: If the value of a numerical variable depends on the outcome of an experiment, we call the variable a random variable . Random Variable A function X : Ω m→ R is called a random variable. Standard notation: Denote random variables by capital letters from the end of the alphabet. Example: Very simple Dartboard If we throw three darts on this board, we are usually interested in the number of times the red area has been hit. This count is a random variable! Also note that P ( red ) = 1 9 on any throw. 1
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Dartboard (continued. ..) More formally: We deFne X to be the function that assigns the number of times that the red area is hit in a sequence of three throws. X ( s ) = k, if s consists of k hits to the red area, and 3 k hits to the gray area. X ( s ) is then an integer between 0 and 3 for every possible sequence of throws of 3 darts. That is, the set of possible values for X ( s ) is { 0 , 1 , 2 , 3 } Notation: To avoid cumbersome notation, we write X = x for the event { ω | ω Ω and X ( ω ) = x } . 2
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Practice with notation Example Suppose, 8 bits are sent through a communication channel. Each bit has a certain probability to be received incorrectly. We are interested in the number of bits that are received incorrectly. Use random variable X to “count” the number of wrong bits received. X assigns a value between 0 and 8 to each sequence in the sample space.
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lec7 - Random Variables Intuitive idea: If the value of a...

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