Exponential distribution with pdf: f(x10) = = exp (-x++), x> B B X>0 0>0 Assume ẞ is known. 1) How do you find the cutoff C* If the test has a size a, i.e., a = sup Plô=c*) 2) How do I find the MME of 0 and show it's an unbiased estimator. If X1 is the complete sufficient stat for 0, how do you find the UMVUE of 0?
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- Weekly rates of return (on an annualized basis) for certain securities over a given period are believed to be normally distributed with a mean of 8.00% and variance of 0.25. Give two values x, and x, such that you are 95% sure that annualized weekly returns will be between the two values.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.An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the chemical is first determinedto be present. At very low concentrations, however, the chemical may fail tobe detected even if it is present. This phenomenon is modeled by assuming thatif the concentration is x, the chemical is detected with probability R(x). Let Ydenote the concentration of a chemical in a soil in which it has been determinedto be present. Show that the density function of Y isg(y) = R(y) f (y)/An instrument is used to measure very small concentrations, X, of a certainchemical in soil samples. Suppose that the values of X in those soils in which thechemical is present is modeled as a random variable with density function f (x).The assay of a soil reports a concentration only if the…
- The random variable x follows a uniform probability distribution in the interval (0,1); the probability of obtaining x values outside this range is zero. What is the probability distribution of y = ln x?A melting point test of n = 10 samples of a binder used in manufacturing a rocket propellant resulted in a sample mean of 154.6 °F. Assume that the melting point is normally distributed with σ = 1.5°F. Is it possible to conclude at 1% significance that the melting point is not 155°F? What is the value of zcalc ? Please report your answer in 3 decimal places.The lifetime of a certain type of TV remote control is given by Y . Suppose Y has approximately exponential distribution with mean 8 years. a) Find the probability that a remote control of this type will last more than 15 years. b) Find the probability that of eight such remote controls at least one will last more than 15 years. c) What should the warranty period for these remote controls be if the manufacturer wants 85% of the remote controls to last beyond the warranty period? d) What is the moment generating function of Y .
- 1. Let X be a Poisson random variable with E[X] = ln2. Calculate E[cosπX]. 2. The number of home runs in a baseball game is assumed to have a Poisson distribution with a mean of 3. As a promotion, Mall A pledges to donate 10,000 dollars to charity for each home run hit up to a maximum of 3. Find the expected amount that the company will donate. Mall B also X dollars for each home run over 3 hits during the game, and X is chosen so that the Mall B's expected donation is the same as the Mall A's. Find X.solveIn random sampling from the exponential distribution, f(x) =1 θθe x− , x > 0, θ> 0, find the maximum likelihood estimator of θ and obtain the asymptotic distribution of this estimatorSuppose that Y has an exponential distribution with mean Beta. Show that 2Y/Beta has a chi square distribution with 2 degrees of freedom.
- A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.The lifetime of a bulb is modeled as a Poisson variable. You have two bulbs typesA and B with expected lifetime 0.25 years and 0.5 years, respectively. When abulb’s life ends, it stops working. You start with new bulb of type A at the start ofthe year. When it stops working, you replace it with a bulb of type B. When itbreaks, you replace with a type A bulb, then a type B bulb, and so on. 1. Find the expected total illumination time (in years), given you do exactly 3bulb replacements.2. Your replacements are now probabilistic. If your current bulb breaks, youreplace it with a bulb of type A with probability pp, and with type B withprobability (1 – pp). Find the expected total illumination time (in years),given you do exactly nn bulb replacements, and start with bulb of type A.Answer for part 2 exists in closed form in terms of n and p.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 <…