Suppose we believe that the life length 'T (in hours) of light bulb is exponentially distributed with parameter à = 0.005. We obtain a sample of 150 bulbs, test them and record their burning time say T, T2, T3.... Tiso and tabulate Test the hypothesis that the data represent a random sample from an exponential distribution with à= 0.005 at level of significance, a = 0.01. it
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- Table 6 shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012. a. Let x represent time in years starting with x=0 for the year 1997. Let y represent the number of seals in thousands. Use logistic regression to fit a model to these data. b. Use the model to predict the seal population for the year 2020. c. To the nearest whole number, what is the limiting value of this model?The operator of a pumping station has observed that demand for waterduring early afternoon hours has an approximately exponential distribution with mean 1000cfs (cubic feet per second).a) Find the probability that the demand will exceed 700 cfs during the early afternoonon a randomly selected day.b) What water-pumping capacity should the station maintain during early afternoons sothat the probability that demand will be below the capacity on a randomly selectedday is 0.995?c) Of the three randomly selected afternoons, what is the probability that on at least twoafternoons the demand will exceed 700 cfs? 2. Let Y1 and Y2 be random variables with joint density functionf(y1, y2) = (6/7(y^2+y1y2/2) 0 < y1 < 1, 0 < y2 < 2,0, elsewherea) Find marginal density functions. Are Y1 and Y2 independent?b) Find P(0 < Y1 < 0.3, −2 < Y2 < 1).c) Find P(0.6 < Y1 < 1|0 < Y2 < 1). 3.The joint density function of Y1 and Y2 is given byf(y1, y2) = (y1 + y2), 0 <…Assume that the time it takes for bacterial growth on a dairy product after it is taken out of fridge follows an Exponential distribution with a mean of 10 minutes. What is the probability that bacterial growth will not happen in 15 minutes if the dairy product has already been out of the fridge for 10 minutes?
- Suppose X1, . . . , Xn ∼ Exponential(λ) is a set of n observations drawn independently from an Exponential distribution. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 1000 cfs (cubic feet per second). a)Of the three randomly selected afternoons, what is the probability that on at least two afternoons the demand will exceed 700 cfs?Suppose that Y has an exponential distribution with mean Beta. Show that 2Y/Beta has a chi square distribution with 2 degrees of freedom.
- The following data consists of the times to relapse and the times to death following relapse of 10 bone marrow transplant patients. In the sample patients 4 and 6 were alive in relapse at the end of the study and patients 7–10 were alive, free of relapse at the end of the study. Suppose the time to relapse had an exponential distribution with hazard rate lambda and the time to death in relapse had a Weibull distribution with parameters theta and alpha (a) Construct the likelihood for the relapse rate lambda. (c) Suppose we were only allowed to observe a patients death time if the patient relapsed. Construct the likelihood for theta and alpha based on this truncated sampleThe operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 1000 cfs (cubic feet per second). b) What water-pumping capacity should the station maintain during early afternoons so that the probability that demand will be below the capacity on a randomly selected day is 0.995? c) Of the three randomly selected afternoons, what is the probability that on at least two afternoons the demand will exceed 700 cfs?The random variable X is exponentially distributed, where X represents the waiting time to see a shooting star during a meteor shower. If X has an average value of 45 seconds, what are the parameters of the exponential distribution?
- The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 100 cfs (cubic feet per second).a) Find the probability that the demand will exceed 200 cfs during the early afternoon on arandomly selected day.b) What water-pumping capacity should the station maintain during early afternoons sothat the probability that demand will exceed capacity on a randomly selected day isonly .01?Suppose X1, . . . , Xn ∼ Exponential(λ) is a set of n observations drawn independently from an Exponential distribution.(a) Write out the likelihood function. (b) Write out the log-likelihood function. (c) Find the score function by taking the partial derivative of the log-likelihood function. (d) Set the score function equal to zero and solve for the parameter λ. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.The survival time of a certain type of mosquito in days following adult emergence is assumed to follow an exponential distribution with parameter λ. We observe 6 mosquitoes, and see lifetimes (4, 12, 6, 19, 9, 5). What is the MLE for λ? Suppose that the data was as above, except that instead of dying after 19 days, the fourth mosquito was observed to be still alive after 20 days, with the exact time of death not known. Now what is the MLE for λ?