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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 λ if its distribution function F is given by: F ( x ) = 1- e- λx 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 light–bulb 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- λx . Notice that the bigger the value of λ , the faster the decay. This indicates that for large λ 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
- Probability theory, Exponential distribution