1. What is the probability that an arrival to an infinite capacity 4 server Poison queueing system with λ/μ = 3 and Po = 1/10 enters the service without waiting?
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- 2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter l. a) If X1, X2, . . . , Xn are the times, in minutes, between successive customers selected randomly, estimate the parameter of the distribution. b) b) The randomly selected 12 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.3 error and 4% risk, find the minimum sample size.6. Consider a birth and death process with birth rates ; = (i + 1)., i > 0, and death rates µ¡ = iµ, i >0. (a) Determine the expected time to go from state 0 to state 4.Consider a linear birth–death process where the individual birth rate is λ=1, the individual death rate is μ= 3, and there is constant immigration into the population according to a Poisson process with rate α. Please explain and show work! (a) State the rate diagram and the generator. (b) Suppose that there are 10 individuals in the population. What is the probability that the population size increases to 11 before it decreases to 9? (c) Suppose that α = 1 and that the population just became extinct. What is the expected time until it becomes extinct again? α in Greek letter alpha.
- Example 17.2 The deviation of the size of an item from the midpoint of the tolerance field of width 2d equals the sum of two random variables X and Y with probability densities and f(x) ❤(y) = = 1 0x V/Z #7 CXP { - 12/0 exp x² 20² √2TT √/2= exp{-2013) · 20² Determine the (conditional) probability density of the random variable X for the nondefective items if the distribution p(y) does not depend on the value assumed by X.29)The actual tracking weight of a stereo cartridge that is set to track at 3 g on a particular changer can be regarded as a continuous rv X with the following pdf.f(x)=k(1-(x-3)²) , 2≤x≤40 otherwise (c) What is the probability that the actual tracking weight is greater than the prescribed weight?P= 0.5 (d) What is the probability that the actual weight is within 0.1 g of the prescribed weight? (Round your answer to four decimal places.)P= 0.1495 (e) What is the probability that the actual weight differs from the prescribed weight by more than 0.45 g? (Round your answer to four decimal places.)P=0.3706B3. A device has two components, and the lifetimes of these components are modelled by random variables Y₁ and Y₂. The first component cannot fail before the second component fails, and the joint density of Y₁ and Y₂ is determined to be f(y₁, y2) = 2e-¹e-92 for 0 0.) I'(a)
- 6. Assume a Rayleigh-fading environment in which the sensitivity level of an RX is given by a signal amplitude rmin, then the probability of outage is Pout = Pr{r < rmin} = cdf (rmin). (b) The fading margin (level of protection against fading) is expressed as M = (30.5) Cmin or (in dB) Мав — Сав — Cmin dB (30.6) %3| dB - Determine a closed-form expression for the required fading margin, in dB, as a function of outage probability Pout-43. For a time series with linear trend i.c. Y = Bo + B₁t + Ct, verify the least squares estimators for ₁ on slide 12.
- 2.Consider the following panel model to examine the effect of retirement on consumption expenditure, consit, of individual i over years t=1,…,3: (B1) log(consit) = β0 + β1retiredit + β2ageit + β3marriedit + β4healthit + δ1Yr2t + δ2Yr3t + ai + uit Where: retiredit is a dummy variable equal to 1 if individual i is retired on year t and 0 otherwise ageit is the individual's age in years marriedit is an indicator variable for whether the individual is married (1) or not (0) in year t healthit is an indicator variable equal to 1 if the individual is in 'good health' and 0 otherwise Yr2 is a dummy variable equal to 1 in year t=2 and 0 otherwise Yr3 is a dummy variable equal to 1 in year t=3 and 0 otherwise Using the information above, answer the following 3 questions. [i] Give two (2) examples of the kind of variables captured by the term ai in Model (B1). [ii] What is the crucial assumption we must make so that the random effects (RE) estimator is consistent? Under this assumption, why is…2
