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Unformatted text preview: A Note on the Exponential Distribution January 15, 2007 The exponential distribution is an example of a continuous distribution. A random variable X is said to follow the exponential distribution with parameter if its distribution function F is given by: F ( x ) = 1- e- x for x > 0. Recall that the distribution function F ( x ) = P ( X x ) by definition and is an increasing function of x . Since F (0) = 0, it follows that X is bigger than 0 with probability 1. The exponential distribution is often used to model the failure time of manufactured items in production lines, say, light bulbs. If X denotes the (random) time to failure of a lightbulb of a particular make, then the exponential distribution postulates that the probability of survival of the bulb decays exponentially fast to be precise, P ( X > x ) = e- x . Notice that the bigger the value of , the faster the decay. This indicates that for large the average time of failure of the bulb is smaller. This is indeed true. It is not difficult to check (verify this) thatbulb is smaller....
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- Fall '09