Let Y be a random variable with an exponential distribution with E(Y) = 0, and supposed a single observation of Y wastaken from the population.a) Show that the pivotal2Yhas a chi-square distribution using the moment generating function technique. What are thedegrees of freedom for this distribution?b) Using the pivotal quantityderive a 95% confidence interval for 0.c) Imagine an observation of Y = 1.5 was collected, compute the 95% confidence interval. Does this interval contain thepopulation parameter? Explain your answer.

From the given information, Y be a random variable with an exponential distribution with E(Y)=θ.…

Answered: Let Y be a random variable with an…

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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Show that if X is a random variable having the Pois-son distribution with the parameter λ and λ→q, then the moment-generating function ofZ = X − λ√λ that is, that of a standardized Poisson random variable, approaches the moment-generating function of the stan-dard normal distribution.
An electronic module used by the Navy in a sonar device requires replacement onthe average once every 16 months and fails according to a Poisson process. Suppose that the Navy places these sonar devices into service on the same date in eightdifferent aircraft carriers. If the modules are replaced immediately after failure andthe budget allows for exactly 40 spares over five years, what is the probability thatthe budget is exceeded? (Hint: Use a normal approximation to the Poisson.) a. The probability that a single carrier sent on a six-month mission will not requirereplacement of the module during that time.
The number of phone calls received at a certain residence in any period of c hours is a Poisson random variable with parameter λ=0.7c. What is the probability p that the phone rings during a given 5 minute period? Give your answer to 4 decimal places. p= How long a period t, in minutes, must one wait for the probability of at least one phone call during that period to be at least 0.5? Give your answer as an integer number of minutes. t=
Suppose small aircraft arrive at a certain airport according to a Poisson processwith rate α= 4 per hour, so that the number of arrivals during a time period of thours is a Poisson random variable with parameter μ = 4t.(a) What is the probability that exactly 6 small aircraft arrive during a 2-hourperiod?(b) What is the probability that at least 6 small aircraft arrive during a 2-hourperiod?(c) What are the expected value and standard deviation of the number of smallaircraft that arrive during a 100-min period?(d) What is the probability that at least 20 small aircraft arrive during a 2.5-hourperiod?(e) What is the probability that at most 20 small aircraft arrive during a 2.5-hourperiod?
Suppose small aircraft arrive at a certain airport accordingto a Poisson process with rate a 5 8 per hour, so thatthe number of arrivals during a time period of t hours isa Poisson rv with parameter m 5 8t.a. What is the probability that exactly 6 small aircraftarrive during a 1-hour period? At least 6? At least 10?b. What are the expected value and standard deviationof the number of small aircraft that arrive during a90-min period?c. What is the probability that at least 20 small aircraftarrive during a 2.5-hour period? That at most10 arrive during this period?
Derive the moment generating function (MGF) of a random variable ?.
An electronic module used by the Navy in a sonar device requires replacement onthe average once every 16 months and fails according to a Poisson process. Suppose that the Navy places these sonar devices into service on the same date in eightdifferent aircraft carriers. If the modules are replaced immediately after failure andthe budget allows for exactly 40 spares over five years, what is the probability thatthe budget is exceeded? (Hint: Use a normal approximation to the Poisson.) c. The expected time to use all 40 spares.?
(a) A structure consists of 100 column footings. Suppose the settlement of eachfooting is exponentially distributed with a mean of 2 inches and the settlementsbetween footings are statistically independent. What is the probability that themaximum settlement Smax among the 100 footings will not exceed 5.0 inches?Compare this exact probability with the answer obtained assuming an appropriateasymptotic distribution for Smax b. What is the probability that the minimum settlement Smax is at least 0.5 inches? c. Determine the mean and coefficient ofvariation of the maximum differential settlement delta(max) defined as (Smax-Smin)
An electronic module used by the Navy in a sonar device requires replacement onthe average once every 16 months and fails according to a Poisson process. Suppose that the Navy places these sonar devices into service on the same date in eightdifferent aircraft carriers. If the modules are replaced immediately after failure andthe budget allows for exactly 40 spares over five years, what is the probability thatthe budget is exceeded? (Hint: Use a normal approximation to the Poisson.) b. The probability that the time of the fifth failure is more than one year after thedevices are placed into service.
An electronic module used by the Navy in a sonar device requires replacement onthe average once every 16 months and fails according to a Poisson process. Suppose that the Navy places these sonar devices into service on the same date in eightdifferent aircraft carriers. If the modules are replaced immediately after failure andthe budget allows for exactly 40 spares over five years, what is the probability thatthe budget is exceeded? (Hint: Use a normal approximation to the Poisson.)

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