ContinuousRan - I NTRODUCTION TO P ROBABILITY T HEORY...

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I NTRODUCTION TO P ROBABILITY T HEORY Continuous Random Variable Continuous Random Variable: if its cumulative distribution function F X ( x ) is a continuous function for all x R . If X is continuous then P ( X = x ) = F X ( x ) - F X ( x - ) = 0 . F X ( x ) = Z x -∞ f X ( t ) dt d dx F X ( x ) = f X ( x )
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I NTRODUCTION TO P ROBABILITY T HEORY P ( a < X b ) = F X ( b ) - F X ( a ) = Z b a f X ( t ) dt. f X ( x ) 0 Z -∞ f X ( x ) dx = 1
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I NTRODUCTION TO P ROBABILITY T HEORY Examples Point Chosen at Random in the Unit Circle: Suppose we select a point at random in the interior of a circle of radius 1 . Let X be the distance of the selected point from the origin. The sample space is C = { ( w, y ) : w 2 + y 2 < 1 } . For 0 < x < 1 , the event { X x } is equivalent to the point lying a circle of radius x . So, P ( X x ) = πx 2 = x 2 .== > Find F X ( x ) and f X ( x ) . P ( 1 4 < X < 1 2 ) = Z 1 2 1 4 2 wdw = 3 16 .
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I NTRODUCTION TO P ROBABILITY T HEORY Let the random variable be the time in seconds between incoming telephone calls at a busy switchboard. Suppose that a reasonable
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