t 1 F t t � 1 F t � 1 1 e � t t 1 1 e λt e � t t e � t e � t 1 F t � P X t

T 1 f t t ? 1 f t ? 1 1 e ? t t 1 1 e λt e ? t t e ?

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Dr. Ammar Sarhan Continuous Random Variables December 7, 2018 29 / 49
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Example 13:The response time X at a certain on-line computer terminal (theelapsed time between the end of a user’s inquiry and the beginning ofthe system’s response to that inquiry) has an exponential distributionwith expected response time equal to 5 sec. Find1probability that the response time is a t most 10 sec? 3Assume that one user is waited for at least 10 sec, what is theprobability that he will get the system’s response within the next5sec?
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Applications of the Exponential Distribution The exponential distribution is frequently used as a model for the occurrence of successive events, such as 1 Customers arrivals at a service facility, 2 Calls coming to a switchboard. The reason for this is that the exponential distribution is closely related to the Poisson process. Proposition Suppose that the number of events occurring in any time interval of length t has a Poisson distribution with parameter α t ( where α is the expected number of events occurring in 1 unit of time ) and the numbers of occurrences in non-overlapping intervals are independent of one another. Then the distribution of elapsed time between the occurrences of two successive events is exponential with parameter λ = α . Dr. Ammar Sarhan Continuous Random Variables December 7, 2018 31 / 49
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Example 14: Exponential, Poisson and binomial Suppose that calls are received at 24-hour hot-line according to aPoisson process with rateα=12per day. Compute:1The probability that more than 2 days elapse between twosuccessive calls.2The expected waiting time between two successive calls.3Given that they have just received a call, what is the probabilitythat they will receive a call within the next 16 hours? 4 Probability that there is 3 calls in 4 days. 5 The expected number of calls in 18 days. 6 Probability that at least 11 calls in 18 days. 7 Probability of a call-free day. Give the answer in 1 decimal place. 8 Probability of no more than 3 call-free days in 10 days. 9 The expected number of call-free days in 10 days. Dr. Ammar Sarhan Continuous Random Variables December 7, 2018 32 / 49
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Dr. Ammar Sarhan
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