(i) On average, Amy receives 10 text messages per hour. Suppose the arrival times of the text messages are independent and exponentially distributed. What is the probability that Amy has to wait more than 8 minutes for the next text message? (ii) Amy woke up at 7 am in morning. She is eagerly waiting for a text message from her new boyfriend. On a usual day, she waits only about 5 minutes for his first message of the day after waking up, but the actual wait time is exponentially distributed. Now it is 7.04 am and there is no message yet... How much longer do you expect her to wait for his message? What is the probability that his message will be received after 7.10 am?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
(i) On average, Amy receives 10 text messages per hour. Suppose the arrival times of the text messages are independent and exponentially distributed. What is the
(ii) Amy woke up at 7 am in morning. She is eagerly waiting for a text message from her new boyfriend. On a usual day, she waits only about 5 minutes for his first message of the day after waking up, but the actual wait time is exponentially distributed. Now it is 7.04 am and there is no message yet... How much longer do you expect her to wait for his message? What is the probability that his message will be received after 7.10 am?
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