Also the increments of nt over disjoint intervals are

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Unformatted text preview: d geometric distributions by showing that the exponential distribution is the limit of scaled geometric distributions. In essence, the exponential 80 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES distribution is the continuous time analog of the geometric distribution. When systems are modeled or simulated, it is useful to be able to approximate continuous variables by discrete ones, because digital computers can only represent quantities to finite precision. Let λ > 0 and let h be a small positive number. We shall define a discrete-type random variable T which is approximately exponentially distributed with parameter λ. Think of T as the lifetime of a lightbulb. Use time clicks of duration h. Let L be the number of clicks until the lightbulb fails. Suppose L is geometrically distributed, with parameter p = hλ. Thus, the pmf of L is given by pL (k ) = (1 − hλ)k−1 hλ, and the interpretation is that the lightbulb fails during each click with probability hλ, given that it didn’t fail previously. So (1 − hλ)k−1 is the probabi...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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