Let X = the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with ? = 1, (which is identical to a standard gamma distribution with ? = 1), compute the following. (If necessary, round your answer to three decimal places.) (d) P(1 ≤ X ≤ 3)

Given an exponential distribution with the notation: X ~ Exp(0.1). Find the following: (the decay parameter (m) = 110110) d) Find the value of the 30th percentile. (Round your answer to two decimal places.)k =

2. Determine whether each of the following distributions are exponential family. For those that are, find a set of complete, minimum sufficient statistics. For those that are not, find a sufficient statistic. (a) Location Exponential: f(x) = e-(-0), x > 0. (b) f(x) = 0(1+x)(1+0), x > 0 (c) logN(u,02): f(x) = 1 a exp{-(log x-μ)²), x > 0. αν2πσ2 },

Use R to complete the following. R code and R output are requested for solution. Please keep your solution clear and easy to read. Unclear solution will not be grade. O for manual solution. Carry out a simulation experiment to illustrate the central limit theorem when the population distribution is arcsine with probability density function f(x) = 1 T√x(1-x) x = (0, 1). Consider the four sample sizes n = 1, n = 3, n = 5, n = 10, and in each case use 10,000 replications. Note: To simulate a random sample of size n from the arcsine distribution in R, use the command rbeta(n,1/2,1/2).