You can use either � which is the average rate or how many successes per time

# You can use either ? which is the average rate or how

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Example 3.14The sirens, while perched on their aesthetically pleasing fjord, were beckoning for Odysseusto come hither. If it on average takes about 1 minute for a captain to navigate his boatstoward the sirens, what is the probability that Odysseus will steer his ship towards themafter 5 minutes?What is the probability that he takes at most 3 minutes?What is theprobability that it takes between 30 and 90 seconds? What is the probability it takes lessthan 300 seconds knowing it took more than 100 seconds?
Example 3.15 Suppose the time it takes a puppy to run and get a ball, say T, follows an exponential distribution with a mean of 30 seconds. State the distribution and parameters of T. What is the probability that it takes the puppy more than 50 seconds to get the ball? Assuming independence, what is the probability that it takes the puppy less than 40 seconds to fetch each of the next 5 balls? What is the probability that it will take the puppy more than 45 seconds to get the ball knowing that it took the puppy longer than 20 seconds? The answers are .1889, . 736403 5 = .2166, and .4346 respectively. Example 3.16 You and 3 friends decide to drive from West Lafayette to Boston to watch the Patriots lose. The duration of a round trip, say D, has an exponential distribution with a rate of 1 trip per 20 hours. Find the following probabilities: D is at most 15 hours, D is between 15 and 25 hours, D exceeds 25 hours, and D is at most 40 given that it is more than 15. Lastly, calculate the mean and variance of D. .5276, .1859, .2865, .7135, 20, and 400 respectively 3.4 Poisson Processes For a specified event that occurs randomly in continuous time, an important application of prob- ability theory is in modeling the number of times such an event occurs. The following are several examples of such random phenomenon. The number of patients that arrive at a hospital emergency room. The number of customers that enter a particular bank. The number of accidents at an intersection. The number of alpha particles emitted by a radioactive substance. Consider an event that occurs randomly and homogenously in continuous time at an average rate of λ per unit of time. We will refer to the occurrence of the event as a success. If we begin counting successes at time 0, and, for each time, t 0, we let N(t) = the number of successes by time t ( t). Automatically, this implies that N(0) is 0. We say such a counting process is a
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