Assume 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
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Assume 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.
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- b and c part Buses arrive at the station according to a Poisson process at a rate of lambda = 0.4 per minute. Assume at the starting condition is time = 0. Imagine when a bus stops, the probability 1 person gets off at the bus station is p = 0.7, and the probability 2 people get off is p = 0.3, independent of everything else. Let X denote the number of people that get off at the bus station in the first 5 minutes. a) Find E[X] b) Calculate P{X = 2} c) Compute Var(X)Find P(X=4) if X has a Poisson distribution such that 3P(X=1)=P(X=2)Let X₁, X₂, ..., Xn and Y₁, Y2, ..., Ym be random samples from populations with moment generating 25 respectively. functions Mx, (t) = e³t+t² and My, (t) = 1-2t, 3 i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X₁ + X5. ii) What is the value of the sample size n, if P[1(X₁ - X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ — µç| ≥ 10) < 0.04?
- Thank youPedestrians 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?b) A random variable X follows an exponential distribution, X~Exp(0), with parameter 0 > 0. Find the cumulative distribution function (CDF) of X. i) ii) Show that the moment generating function (MGF) of X is M(t) = iii) Use the MGF in (ii) to find the mean and variance of X.
- Suppose that X and Y are independent, X~B(10, 0.5), Y follows Poisson distribution with parameter 4. Which of the following statement is not true? A.E(X+Y)=9 B.E(X-Y)=1 C.D(X+Y)=6.5 D.D(X-Y)=-1.5Suppose the customers arrive at a Poisson rate of on eper every 12 minutes, and that the service time is exponential at a rate of one service per 8 minutes. What are the average number of customers in the system(L) and the average time a customer spends in the system(W)?Use the geometric distribution to derive E(Y), E(Y^2), and V(Y) from the Poisson distribution
- 7. (Sec. 3.6) Consider sending packets of data through the internet, and we have a counter that determines the number of missing/corrupted packets. Suppose that number X has a Poisson distribution with parameter 23. = (a) What is the probability that the counter determines that the transmission has exactly one miss- ing/corrupted packet of data? (b) What is the probability that the counter determines that the transmission has at least 3 miss ing/corrupted packets of data? (c) If two transmissions of data across the internet ar that at least one of them does not contain a missing/corrupted packet? e independently selected, what is the probabilityLet P be a random variable having a uniform distribution with minimum 0 and maximum 3 i.e. P ~ Uniform(0, 3). Let Q = log (P/(3-P)). Find E[P]. You areexpected to solve this problem without using Method of Transformation.Let X has Poisson distribution with parameter 2 such that P(X=2) =9P(X=4) +90P(X=6). Find the value of parameter