Prove that - 1. The waiting time for a Poisson distribution is Exponentially distributed. i.e. For a Poisson process with parameter 2, the time for a new arrival is
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Q: Prove that - 1. The waiting time for a Poisson distribution is Exponentially distributed. i.e. For a…
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A: True
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- In a video game with a time slot of fixed length T, signals are generated according to a Poisson process with rate λ, where T > 1/λ. During the time slot you can push a button only once. You win if at least one signal occurs in the time slot and you push the button at the occurrence of the last signal. Your strategy is to let pass a fixed time s with 0 < s < T and push the button upon the first occurrence of a signal (if any) after time s. What is your probability of winning the game? What value of s maximizes this probability?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 Distribution2a) The number of flowers per square meter in Sarah’s garden has a Poisson distribution with mean 0.35. Her garden is covered with 150 square meters of grass. Find lambda λ? 2b) The number of flowers per square meter in Sarah’s garden has a Poisson distribution with mean 0.35. Her garden is covered with 150 square meters of grass. Using Normal approximation, we will need to find the probability that the Sarah’s garden will contain less than 45 flowers. First graph and answer what is the continuity correction? 2c) Using the previous results for lambda and continuity correction, find z, then graph and use your table to find φ table value of z Write down your final answer for the probability that Sarah’s garden will contain less than 45 flowers as a decimal number with 4 decimal places.
- Q1) A document printing machine receives documents according to a Poisson process with an expected interarrival time of 5 minutes from the legal department. When the machine has just one document to print, the expected processing time is 5 minutes. When machine has more than one document, then it does not require initial set-up time and expected processing time for each document to 4 minutes. In both cases, the processing times have an exponential distribution. (a) Construct the rate diagram for this queueing system. (b) Check that this is a birth-and-death process and find the steady-state distribution of the number of documents in the system but not yet finished printing. (Hint: Use the sum of Geometric series) (c) Derive L and Lq for this system. (Hint: Use the sum of Arithmetic-Geometric series) (d) Use the information of L and Lq to determine W and Wq. Q2) Jacob runs a shoe repair store by himself. Customers arrive to bring a pair of shoes to be repaired according to a Poisson…In a certain desert, the probability that it rains on any given day is 1/500 (i.e., 0.2 percent). Assume that whether it rains or not on a given day does not depend on what happens on other days(a) Write a formula for the probability that it will rain exactly twice during the next 1000 days.(b) Use the Poisson Limit Theorem to approximate this probability.Prove the following property of the compound Poisson process:1. it has independent and stationary increments.
- Suppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?At 15:00 it is the end of the school day, and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average, 24 students leave the school every minute. (e) There are 200 days in a school year. Given that Y denotes the number of days in the year that at least 700 students leave before 15:30, find (ii) P(Y > 150).In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.
- Let X have a Poisson distribution with parameter λ. Show that E(X)=λ directly from the definition of expected value. (Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out λ and show that what is left sums to 1.)Consider the following techniques of transforming time series data. Choose all that applyAssume that the variables Y1, Y2,... in a compound Poisson process have Bernoulli distribution with parameter p . Show that the process reduces to the Poisson process of parameter λp.