prove that the waiting time for a Poisson distribution is Exponentially distributed and an Exponential distribution can be obtained as a limit of a Geometric Distribution
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1). prove that the waiting time for a Poisson distribution is Exponentially distributed and an Exponential distribution can be obtained as a limit of a Geometric Distribution
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- Assume that we collect data (time how late the bus was at the end of the day) in related to following problem. A bus is scheduled to stop at a certain stop every half hour. At the end of the day due to delays that often occur earlier in the day, the bus is likely to be late. The director of the bus line claims that the length of time a bus is late at the end of the day is uniformly distributed and the maximum time that a bus is late is 20 minutes. Assume the lateness of the bus on different days are independent. (a) In a randomly selected day, what is the probability this bus is late more than 5 minutes at the end of the day? If we consider consecutive days, what is the probability that the 6th day will be first (b) day this bus is late more than 5 minutes at the end of the day? What is the probability that at least 10 consecutive days there, before a day with this bus is late more than 5 minutes at the end of the day? (d) . probability that less than 60 of them will be more than 5…Determine if the following scenarios follow a Poisson distribution or do not follow a Poisson Distribution. • You work in a shoe shop and you want to find the distribution of the total number of people who come into the store in a day. You also observe that there are usually more people who come into the shop over lunchtime (10-2 pm) and more people who come into the store after they finish work (5 pm-6 pm). Let X be the number of people who come into the shoe shop on a given day. No - not Poisson • You love peanut butter and want to know how many people buy peanut butter from Trader Joe's. One summer you have nothing to do so you go and stand in Trader Joe's every day and count how many people buy peanut butter in a given week. You assume that consumers do not impact each other's decisions. Let Y be the number of people who buy peanut butter in a given week. Yes - Poisson You want to know how many people in a group of 100 take painkillers if they have a headache. You think that the…Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least 95% of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion? Answer the previous question for a random day during the year.
- The time intervals between successive barges passing a certain point on a busy waterway have an exponential distribution with mean 12 minutes. (a) Find the probability that the time interval between two successive barges is more than 8 minutes. (b) Find a time intervalt such that we can be 90% sure that the time interval between two successive barges will be greater than t. Loount ▼Yhmeer is watching a shower of meteors (shooting stars). During the shower, he sees meteors at an average rate of 1.3 per minute. (a) State the conditions required for a Poisson distribution to be a suitable model for the number of meteors which Yhmeer sees during a randomly selected minute (b) Using Excel or otherwise, determine the probability that, during one minute, Yhmeer sees (i) exactly one meteor (ii) at least 4 meteorsA study prospectively examined whether sleep-disordered breathing was associated with an increased risk of death from any cause in a cohort of 600 adults participating in the Sleep Heart Health Study. Study participants were classified into four groups depending on the extent of their sleep-disordered breathing (none, mild, moderate, or severe). The counts of deaths over the course of the study are reported for each group in the following two-way table. Mild Severe 100 Death No Death Total None 40 100 140 40 100 140 Moderate 20 100 120 Total 200 400 600 100 200 We want to know whether the study findings give evidence of a significant difference of the number of deaths between the different groups. Set up the alternative hypothesis (H, : The distribution of the categorical variable is not as given by the null hypothesis (lack of fit) At least ones of the means is different from the others O The two categorical variables are dependent The distribution of the categorical variable is not…
- The time intervals between successive barges passing a certain point on a busy waterway have an exponential distribution with mean 10 minutes. (a) Find the probability that the time interval between two successive barges is less than 8 minutes. (b) Find a time interval t such that we can be 95% sure that the time interval between two successive barges will be greater than t.Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 250 numerical entries from the file and r = 60 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1. Test the claim that p is less than 0.301 by using α = 0.01. What does the area of the sampling distribution corresponding to your P-value look like? a. The area in the right tail of the standard normal curve. b. The area not including the right tail of the standard normal curve.…If a number is chosen from a set of three-digit numbers, what is the probability that the number is divisible by 5?
