1. (6 points) Let X1​,…,Xn​ be an i.i.d. sample from the exponential population with mean 1 . For an unknown θ and for i

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1. (6 points) Let X1​,…,Xn​ be an i.i.d. sample from the exponential population with mean 1 . For an unknown θ and for i=1,2,…,n, define Yi​=θe−Xi​. (a) Determine the distribution of Y1​. (b) Define U=Y(1)​ and V=Y(n)​ as the smallest and largest order statistics for Y1​,…,Yn​, respectively. Determine the joint density function of (U,V). (c) Find the marginal distribution of V. In addition, determine the limiting distribution of (V/θ)n. (d) Find the mean and variance of V. (e) Using the sample Y1​,…,Yn​, find a sufficient statistic of θ. (f) Is the statistic in part (e) complete? Prove or disprove. If it is not a complete statistic, find a complete statistic. 2. (4 points) Let X1​,…,Xn​ be a random sample from the population with density f(x∣θ)=θx(1−θ)1−x, for x=0,1, and 0<θ<1. (a) Does this population come from an exponential family? Explain. (b) Determine the UMVUE of θ. (c) Does the UMVUE of θ reaches the CRLB? (d) Show that the UMVUE of θ2 is T=Xˉn−1nXˉ−1​. 3. (4 points) Let X1​,X2​,…,Xn​ be a random sample taken from the distribution with p.d.f. f(x∣θ)=θ1​x1/θ−1, for 0<x<1, and θ>0. (a) Determine a method-of-moments estimator for θ. (b) Use the Likelihood-Ratio-Test (LRT) method to determine a size α test of H0​:θ=θ0​ versus H1​:θ=θ0​, for θ0​>0. Express your decision region(s) using chi-square value(s). (c) Convert the above test to a two-sided 100(1−α)% confidence interval for θ. (d) Determine the UMP test, at level α, for H0​:θ=1 versus H1​:θ>1 and express the nulldistribution of the test statistic as a chi-square random variable. 4. (4 points) A discrete random variable X with properties P(X=x)=a(x)C(θ)θx​, for x=0,1,…;a(x)≥0;θ>0 is said to have a power series distribution. (a) Show that the moment generating function of X is MX​(t)=C(θ)C(θet)​. (b) Show that the binomial and Poisson distributions are special cases of the power series distribution and determine θ and C(θ) for both cases. (c) Show that the mean of X is E(X)=θdθd​lnC(θ). (d) Suppose that X1​,X2​,…,Xn​ are i.i.d. random variables according to a power series distribution. Show that T=i=1∑n​Xi​ is sufficient for θ.