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# post13 - .75 e –.075 = 1.078 –.472.928 =.491 4.2...

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1 STAT 511-2 Spring 2012 Lecture 13 Feb 8, 2012 Jun Xie Finish contents from the last post, Poisson distribution and pdf of continuous rv. 4.1 Probability density function (continued) Definition A continuous rv X is said to have a uniform distribution on the interval [ A , B ] if the pdf of X is Example 5 “Time headway” in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let X = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. Assume the pdf of X It’s easy to check that f ( x ) 0 and 𝑓 ? 𝑑? −∞ =1. The probability that headway time is at most 5 sec is = e .075 ( e .75

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Unformatted text preview: .75 + e – .075 ) = 1.078( – .472 + .928) = .491. 4.2 Cumulative distribution functions and expected values The cumulative distribution function Definition The cumulative distribution function F ( x ) for a continuous rv X is defined for every number x by F ( x ) = P ( X x ) = °±?²³? ? −∞ If X is uniform on [A,B], its cdf is 2 Proposition Let X be a continuous rv with pdf f ( x ) and cdf F ( x ). Then for any number a , P ( X > a ) = 1 – F ( a ) and for any two numbers a and b with a < b , P ( a X b ) = F ( b ) – F ( a ). Example 7 Proposition If X is a continuous rv with pdf f ( x ) and cdf F ( x ), then at every x at which the derivative F ( x ) exists, F ( x ) = f ( x )....
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