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Could you solve (iv)?
Thank you.
![Q4 Consider a Poisson process with rate A. Compute
Q4(i.) Expected time of the 1Oth event.
Q4(ii.) Probability that the 10th event occurs 2 or more time units after the 9th event.
Q4(iii.) Probability that the 10th event occurs later than time 20.
Q4 (iv.) Probability that 2 events in the time interval [1, 4] and 3 events in the time interval [3, 5].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2F05585b66-1860-42a6-ba2a-bb1fa26d4b0b%2F3s6hwcv_processed.png&w=3840&q=75)
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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?Let {N_1(t)} and {N_2(t)} be two independent Poisson processes with rates λ1=1 and λ2=2, respectively. Find the probability that the second arrival in N_1(t) occurs before the third arrival in N_2(t). Round answer to 4 decimals.The number of requests for assistance received by a towing service is a poisson process with rate α = 2 per hour. After realizing that he did not miss any calls for assistance when taking a 15 minute break, the operator decides to take another 15-minute break. What is the probability that he will not miss any calls again? We want to determine P[no calls in time interval (0 , 1/2)│no calls in time interval (0 , 1/4)], where the unit used for time interval is hours.
- Pedestrians approach a crossing from the left and right sides following independent Poisson processes withaverage arrival rates of λL = 5 and λR = 1 arrivals per minute. Each pedestrian then waits until a lightis flashed, at which time all waiting pedestrians must cross to the opposite side (either from left to right orfrom right to left). Assume that the left and right arrival processes are independent, that the light flashesevery T = 2 minutes, and that crossing takes zero time – it is instantaneous.1. What is the probability that in a particular crossing, there are total 10 pedestrian and they are allcrossing from left to right?Suppose taxicab arrivals from 3 to 5 p.m. are a Poisson process. If the cabs arrive at an average rate of 18 per hour, what is the probability that a person will have to wait up to 5 minutes for a cab?A traffic light is red for 20 seconds and green for 40 seconds. Cars arrive at the traffic light as a Poisson process with rate 2 per minute. They take exactly 20 seconds to go through the light. A car cannot start going through until the previous one has cleared the light. Assuming, as an initial condition, that there are no cars at the traffic light when it turns red: a. What is the probability that at least one car will arrive when the light is still red?b. What is the probability that there will be no cars at the light when the light next turns red?
- 2a) 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.Consider a Poisson process with parameter λ = 0.8. (a) What is the probability that the time between two adjacent events is longer than 1.5? (b) What is the probability that there will be at least three events in a period of length 2?If it is known that the random process X (t) is a stationary process in a broad sense, could the following autocorrelation function belong to this process? Explain the reason for your answer.
- A pump operates 1000 hours/year. Under a minimal repair concept, the pump failures generated a non-homogenous Poisson process having the following intensity function with t measured in operating hours. Row(t)=0.00003t^2. a) From the information given, is the rate of occurance of failure (ROCOF) increasing, decreasing or remaining constant? b) Calculate the number of expected failures of the pump over 1000 hours of operation. c) Calculate the MTBF for the 1000-hour operation. d) The repair time of the pump is best described by the following probability density function h(t)=t^2/3 for 0<_t<_3 hours. WHat is the mean time of repair, in hours? e)What is the inherent availability of the pump over the 1000 hours?This is a Stochastic Process question. Each customer makes claims following a Poisson process with rate 0.44 per year.For Scheme A, there are 3 discount levels: 0%, 10% and 20%.For a customer in Scheme A, he or she will move between discount levels throughout the year at continuous times.The annual premium to be paid at the start of each year is determined by the discount level he or she is in at that point of time, irrespective of the movements between the discount levels throughout the year.If a customer in Scheme A makes a claim, he or she will immediately move to a worse discount level, if possible, or remain in the 0% level.Independently of the customer making claims, for each customer in Scheme A independently the insurance company will perform a 'bump' where the customer will move to a better discount level if possible (or remain iin 20% otherwise) every so often as a customer loyalty programme with time between bumps exponentially distributed with mean 2 years. Find the…Let W1, W2, . . . be an uncorrelated random sequence with mean 0 and variance 1.Define the discrete-time random process {Xn : n ∈ N} := {X1, X2, . . .} by Xn = aXn−1 + Wn(n ∈ N) with a and X0 given. For each of the following two separate cases, find the mean functionmX (n) (n ∈ N) and covariance function CX (m, n) (m, n ∈ N) for the process {Xn : n ∈ N}, anddetermine if it is wide-sense stationary.a. a = 1 and X0 = 0.b. |a| < 1 and X0 is a random variable with mean 0 and variance 1/(1 − a2), uncorrelated withW1, W2, . . . .
