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 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? 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 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 Dr. Ammar Sarhan  • • • 