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Unformatted text preview: Exponential Distribution This distribution is commonly used to model waiting times between occurrences of rare events, lifetimes of electrical or mechanical devices. Exponential density A random variable X has exponential density if f X ( x ) = braceleftbigg λe λx if x ≥ otherwise λ is called the rate parameter . We say that the random variable X ∼ Exp ( λ ) Mean, variance and distribution function are easy to compute. They are: E [ X ] = 1 λ V ar [ X ] = 1 λ 2 Exp λ ( t ) = F X ( t ) = braceleftbigg if x < 1 e λx if x ≥ 1 Density functions of exponential variables for different rate parameters 0.5, 1, and 2. f 0.5 f 1 f 2 x 2 Example 4.5 (Baron) Example: Jobs are sent to a printer at an average of 3 jobs per hour. (a) What is the expected time between jobs? (b) What is the probability that the next job is sent within 5 minutes? Solution: Job arrivals represent rare events, thus the time T between them is Exponential with rate 3 jobs/hour i.e. λ = 3 . (a) Thus E ( T ) = 1 /λ = 1 / 3 hours or 20 minutes. (b) Using the same units (hours) we have 5 min.=1/12 hours. Thus we compute P ( T < 1 / 12) = Exp 3 (1 / 12) = 1 e 3 · 1 12 = 1 e 1 4 = 0 . 2212 3 Exponential Distribution: Example Note: The following example will be continued throughout the remainder of this class. Example: Hits on a webpage Suppose we are told that, on average, there are 2 hits per minute on a specific web page....
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 Spring '11
 Zhou
 Probability theory, Exponential distribution, CDF, Baron

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