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- If Y has a binomial distribution with n trials and a probability of success p, show that the functiongenerator of moments for Y is: m(t) = (pe' +q)n with q = 1− pIf X follows a binomial distribution with p = 0.1 and n =100, find the approximate value of P(2 ≤ X ≤ 4) using the normal approximation. the Poisson approximation.For the exponential distribution with parameter 3, what is P(1≤X≤2)?
- If X1 and X2 constitute a random sample of size n = 2from an exponential population, find the efficiency of 2Y1relative to X, where Y1 is the first order statistic and 2Y1and X are both unbiased estimators of the parameterSuppose X1,...,Xn are iid with PDF f(x | θ) = 2x/θ2, 0 < x < θ. (a) Write out the joint PDF f(x | θ), and write the likelihood function L(θ | x). (b) Find the MLE for θ. (c) Find the MLE for the median of the distribution.