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MIE360 13 Memoryless - MIE360 Computer Modeling and...

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MIE360 Computer Modeling and Simulation Lecture Notes Daniel Frances © 2010 1 Lecture 16. Special role of Exponential Distribution. It is the only distribution that satisfies a very important property: the forgetful or memory-less property: Expon(0.50) 0 0.5 1 1.5 2 2.5 0 1 2 3 4 Suppose someone were to tell you that no matter how long you had waited for a bus, the probability distribution of the time for the arrival of the next bus remained the same. What would be your first reaction? And you are usually right. Say that the schedule calls for a bus to come in about 5 minutes, with some normally distributed margin of error. Normal(5.00,1.00) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 4 6 8 10 Suppose that 3 minutes had lapsed and you wanted to update the distribution to see what it would look like now. P(bus in minute t | no arrival in [0,3] ) = P ( bus in minute t and no arrival in [0,3] ) / P ( no arrival in [0,3]) Normal(5.00,1.00) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 2 4 6 8 10 Therefore new distribution will be a truncated normal distribution, adjusted to make sure the area under the distribution is still one.
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MIE360 Computer Modeling and Simulation Lecture Notes Daniel Frances © 2010 2 What if the distribution of the time to the next bus was exponentially distributed? Expon(5.00) 0 0.05 0.1 0.15 0.2 0.25 0 10 20 30 40 Again P(bus in minute t | no arrival in [0,3] ) = P ( bus in minute t and no arrival in [0,3] ) / P ( no arrival in [0,3]) And so again the probability distribution will be a truncated exponential distribution, adjusted for the area under the curve to be one. But wait, if we truncate this distribution, it certainly looks very similar to the original. Let’s check it out. Let t 0 be the time of arrival of a bus measured from my arrival at the station and pdf 0 (t 0 ) be the corresponding probability density function. Suppose that after 3 minutes there is still no arrival, and now t
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