Exercise 5 (10pts)Demands on a first aid facility in a certain location occur according to a nonhomogeneousPoisson process having the rate function2tfor 0≤t<1λ(t)= 2for 1≤t<24-t for 2≤t≤4where t is measured in hours from the opening time of the facility.What is the probability that two demands occur in the first two hours of operation?

The objective of the question is to find the probability that two demands occur in the first two…

Answered: Exercise 5 (10pts) Demands on a first…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A model for normal human body temperature, X, when measured orally in °F, is that it is normally distributed, X ~ N (98.2,0.5184). (i) According to the model, what proportion of people have a normal body temperature of 99 °F or more? (ii) Find the normal body temperature such that, according to the model, only 10% of people have a lower normal body temperature. (iii) Let W denote normal human body temperature, when measured orally in °C. Given that W = (X - 32) and X ~ N(98.2,0.5184), what is the distribution of W? (iv) According to the model that you just derived for W, what proportion of people have a normal body temperature of between 36 °C and 36.8 °C?
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 model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.
A Nail salon opens at 10am. During the busy Christmas period people start arriving at 9 am and queue in front of the salon to wait for the salon opening. The arrivals follow a non-homogeneous Poisson process with rate function A(t) = 25t^2. What the average waiting time of a person who is present at the nail salon at 10 am.
Let {N(t); t ≥ 0} be a renewal process with the gamma distributed inter- occurrence times with parameters 2 and 3 .The probability distribution of waiting time until the fifteenth renewal occurs is Select one: a normal distribution with paramters 10 and variance 3.33 O b. normal distribution with mean10 and variance 50 gamma distribution with parameters 30 and 3 O d. gamma distribution with parameters 30 and 45 O C.
Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?
In a continuous-time surplus model, the claim severity is distributed as BN(2, 0.4). Determine the Lundberg upper bound for the probability of ultimate ruin if the initial surplus is 2 and the prIn a continuous-time surplus model, the claim severity is distributed as BN(2, 0.4). Determine the Lundberg upper bound for the probability of ultimate ruin if the initial surplus is 2 and the premium loading factor is 0.25.emium loading factor is 0.25.
#5. Two dice are rolled 300 times. Let X = number of double sixes obtained (out of 300 rolls). (a) Find P(X = 8) using the Poisson approximation to the binomial. (b) Find the mgf of W = 3X + 2.
Suppose you are fitting a survival model and the outcome of interest is time to the diagnosis of cancer (T). A subject entered the study at t=0 and the diagnosis of cancer did not occur by the end of the study period, what kind of event is that? It is an interval-censored observation because an event happened between time of entry and the exit. It is a right-censored observation because no event will be observed during the study period. It is a left-censored observation where the event of interest happened before the study starts. It is a right-censored observation because the event has already occurred.
Exercise 2. A r.v. X is Poisson distributed with parameter λ = 5.5. (a) Calculate the probability that X = 3. (b) Calculate the probability that X < 2. (c) What is the most likely (i.e. highest probability) value for X ? (You can make a graph of f(x) to find the answer).
The arrival of customers to a small local food restaurant may be modeled by a Poisson process with rate of 1 per 15 min period. Assume customer will only arrive if there is at least one customer present. Each customer on average stays for a time exponentially distributed with mean 20 minutes. This is modeled as a birth-death process. (i) Compute the probability that the capacity of 20 will be reached if we start with 5 customers.
A biased coin with probability q of landing heads is repeatedly tossed until the first head is seen. The number of tails X before the first head is modelled as a geometric distribution with probability mass function P(X = x) = g(1-q). The experiment was repeated n times and x1,x2,..., Xn tails were observed. (a) Write down the likelihood for q. Show that the maximum likelihood estimate for q is â = n n+S n where S = x₁. (b) Find the Fisher information and hence the asymptotic variance for â. (c) A Beta(ao,Bo) distribution is chosen as the prior distribution for q. Show that the posterior distribution is Beta(a₁,₁), where you should determine a₁ and ₁. (d) We have n = 5 and observed data x₁,...,x₂ = 4,2,5,6,3. (i) What is the maximum likelihood estimate â? (ii) Find an approximate 95% confidence interval for q. (iii) Before seeing the data, our probability distribution for q has mean 0.4 and standard deviation 0.2. Find values of ao and Bo corresponding to this belief. What is then the…

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