86. In good years, storms occur according to a Poisson process with rate 3per unit time, while in other years they occur according to a Poissonprocess with rate 5 unit time. Suppose next year will be a good yearperwith probability 0.3. Let N(t) denote the number of storms during thefirst time units of next year.(a) Find P{N(t) = n}.(b) Is {N(t)} a Poisson process?(c) Does {N(t)) have stationary increments? Why or why not?(d) Does it have independent increments? Why or why not?(e) If next year starts off with three storms by time t = 1, what is theconditional probability it is a good year?

Answered: 86. In good years, storms occur…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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3. For a time series with linear trend i.c. Y = Bo + B₁t + Ct, verify the least squares estimators for ₁ on slide 12.
Suppose that the login times to a computer network follows a Poisson process with an average of 4 logins per minute. Find the probability of exactly 6 logins in 2 minutes.
Example 9-53. (a) Find the p.d.f. of Xr) in a random sample of size n from the exponential distribution: f(x) = a e i, a>0, x 20. (b) Show that X) and W, = X9-X¢», r < s, are independently distributed. %3D
1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
2
Give that A ∼ Poisson(3) Find P(A > 2)
If S = {a,b,c} with P(a) = P(b) and P(c) = 0.24, find P(a). (Enter answer as a decimal with at least 2 correct decimal places)
Assume 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 ba
77. A system consists of five identical components connected in series as shown: 1 3 4 5 As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with 2 = .01 and that components fail independently of one another. Define events A; : at least t hours }, i = 1, ... , 5, so that the A;'s are independent events. Let X = the time at which the system fails- that is, the shortest (minimum) lifetime among the five components. a. The event {X > t} is equivalent to what event involving A1, ..., As? b. Using the independence of the five A;'s, com- pute P(X > t). Then obtain F(t) = P(X < 1) and the pdf of X. What type of distribution {ith component lasts does X have? c. Suppose there are n components, each having exponential lifetime with parameter å. What type of distribution does X have?
1 163
7. Prove or disprove that the following function is a cumulative distribution function (cdf). If the function is a cdf, is the random variable continuous or discrete? Why? (Note: This is a special case of a logistic distribution.) Fx(x)= 1 1+ e-z -∞<<∞.
7. In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois- son distribution with probability mass function f(x), where f(x) = 0" e- /x! and 0 is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was observed that ng quadrats had x bacteria (x = 0,1, ., c) and that m quadrats had more than c bacteria. Show that the maximum likelihood estimate of 0 is given by -1 mf(c) 1- F(c)] = s In – m - where s = Ecn, and F(c) =E{(x). x=1 x=0

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