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- 3. A random variable X has a cumulative distribution function if 0 0.5). (e) Find P(X < 1.25). (f) Find P(X = 1.25).1.1 Consider a time series process given by (1-B)x, =(1-6,B)z,, where the sequence {z, ) is a white noise with mean zero and variance o. Determine the first four weights of process x, if an ARMA (1,1) model fitted to the data gave =0.8 and 9 = 0.34 (4) Consider an AR(2) model: x=(x)+(x2-x)+2,, where z, - wn(0,0²). 1.2 QUESTION 1 Show that = A(-A) and = 1-A using LSE method.1. Let X be a Poisson random variable on the non-negative integers with rate λ = 4. Let W = 2X + 10. (a) What is the range of W? (b) Find a formula for Pw(k).
- 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?7. a) Suppose that X is a uniform continuous random variable where 0Assume that a time-series plot takes the form of that shown in the following graph: 80 60 60 40 40 20 20 * -20 ** * -40 0 2 4 6 8 10 12 Given this plot, which of the following models would likely give the best fit? O ^ = bo + bit + by + bị t lý = bo + bit + bit ŷ = bob₁ + b₁t On =bo+ bạt + biết biết ba5. Suppose a random sample X1, X2, ..., X, of size n is taken from the distribution with p.d.f f(x;0) = -x1-9)/0, 08. Suppose X₁, XX-f(x;0) with f(x,0)= Find the MLE of the odds of success 9 = ( x ) 0² (₁-0) ² + ₁ x 0 1-0 0),x=0,1,2,...,k;0<0<1.#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 that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?In time-series decomposition, seasonal factors are calculated by Multiple Choice O O O O O SFt (Y) (CMA). SFt= Y/CMAt (CMA+) x (SFt) =Yt. SFt = Yt - CMAt. None of the options are correct.49. Cars arrive at the beginning of a long road in a Poisson stream of rate from time t = 0 onwards. A car has a fixed velocity V> 0 which is a random variable. The velocities of cars are independent and identically distributed, and independent of the arrival process. Cars can overtake each other freely. Show that the number of cars on the first x miles of the road at time t has the Poisson distribution with parameter λE[V-¹ min {x, Vt}].SEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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