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Probability And Statistical Inference (10th Edition)
- A company has 8000 arrivals of Internet traffic over a period of 17,460 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= μx•e−μ x! to find the probability of exactly 3 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?arrow_forwardA company has 9000 arrivals of Internet traffic over a period of 18,050 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)= (μ^x • e^−μ) / x! to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?arrow_forwardFor a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.arrow_forward
- A company has 9000 arrivals of Internet traffic over a period of 20,740 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use the formula P(x)=μx•e−μx! to find the probability of exactly 3 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?arrow_forwardSuppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]arrow_forwardLet X1, X2, …, Xn be a random sample from the Normal distribution N() (a) Using method of moments to estimate the parameters and . (b) Are those estimators unbiased?arrow_forward
- The number of bacteria colonies of a certain type in samples of polluted water has a Poisson distribution with a mean of 2 per cubic centimeter (cm3) a. If four 1-cm3 samples are independently selected from this water, find the probability that at least one sample will contain one or more bacteria colonies. b. How many 1-cm3 samples should be selected in order to have a probability of approximately 0.95 of seeing at least one bacteria colony?arrow_forwardThe number of people arriving per hour at the emergency room (ER) of a local hospital seeking medical attention can be modeled by the Poisson distribution, with a mean of 15 people per hour.The inter-arrival time, X, is defined as the time that passes between successive arrivals of patients seeking medical attention. (a) It has been 4 minutes since the last person seeking medical attention arrived at the ER. What is the probability that at least 9 minutes (in total) will pass until the next medical-attention-seeking person passes through the ER doors? Use four decimals in your answer.arrow_forwardif Y1 and Y2 are independent Poisson random variable with parameters λ1 and λ2 re-spectively. Then find the conditional distribution of Y1 given Y1 + Y2 = n. That is calculateP(Y1 = k|Y1 + Y2 = n).arrow_forward
- Let Yn denote the nth order statistic of a random sample of size n froma distribution of the continuous type. Find the smallest value of n for which theinequality P(ξ0.9 < Yn) ≥ 0.75 is true.arrow_forwardA new printing machine is tested about the number of printing errors per 5m2 paper. It is assumed that the number of printing errors will have Poisson distribution with parameter l. a) If X1, X2, . . . , Xn is a random sample of size n, estimate the parameter of the distribution. b) The randomly selected 12 papers are inspected, and the number of printing errors are found as 2, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, and 0. Estimate the mean number of printing errors, 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.arrow_forwardFind the sampling distributions of Y1 and Yn for ran-dom samples of size n from a population having the beta distribution with α = 3 and β = 2.arrow_forward
- A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON