Let X1, ..., Xn denote an independent random sample from a population with a Poisson distribution...
Let X1...Xn be independent, identically distributed random sample from a poisson distribution with mean theta. a. Find the meximum liklihood estimator of theta, thetahat b. find the large sample distribution for (sqrt(n))*(thetahat-theta) c. Construct a large sample confidence interval for P(X=k; theta)
Q6: Let X1, ..., Xn be a random sample of size n from an exponential distribution, Xi ~ EXP(1,n). A test of Ho : n = no versus Hain > no is desired, based on X1:n. (a) Find a critical region of size a of the form {X1:n > c}. (b) Derive the power function for the test of (a).
Can anyone help me with this problem? Thank you! 7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1: θ θο. (a) For a sample of size n, find a uniformly most powerful (UMP) size-a test if such exists. (b) Take n-?, θ0-1, and α-.05, and sketch the power function of the UMP test. 7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1:...
Let X1,... , Xn be a random sample from the Pareto distribution with pdf { f (r0)= 0, where 0>0 is unknown (a) Find a uniformly most powerful (UMP) test of size a for testing Ho 0< 0 versus where 0o>0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) (b) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = log Xi. (Use quantiles...
Let X1, X2, ..., Xn denote a random sample of size n from a population whose density fucntion is given by 383x-4 f S x f(x) = 0 elsewhere where ß > 0 is unknown. Consider the estimator ß = min(X1, X2, ...,Xn). Derive the bias of the estimator ß.
Let X1, X2, ...,Xn be a random sample of size n from a Poisson distribution with mean 2. Consider a1 = *1782 and în = X. Find RE(21, 22) for n = 25 and interpret the meaning of the RE in the context of this question.
5. (10 points) Let X1,... , Xio be a random sample of size 10 from a Poisson distribution with mean θ. The rejection region for testing Ho :-0.1 vs. 1.1: θ-0.5 is given by Σ"i z > 4. Determine the significance level α and the power of the test at θ : 05. 5. (10 points) Let X1,... , Xio be a random sample of size 10 from a Poisson distribution with mean θ. The rejection region for testing Ho...
Let X1, ..., Xn be a random sample from Gamma(1,41) distribution and Y1, ..., Ym be a random sample from Gamma(1,12) distribution. Also assume that X’s are independent of Y's. (1) Formulate the LRT for testing Ho : 11 = 12 v.s. Hy : 11 + 12; (10 points) (2) Show that the test in part (1) can be based on the following statistic (7 points) T = 21-1 Xi Dizi Xi + [2Y; = (3) Find the distribution of...
Let X1, , Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β-A-Derive the Score test for testing Ho- Let X1, , Xn be a random sample gamma(α, β), assume a is known. Consider testing Ho : β-A-Derive the Score test for testing Ho-
Let X1, X2, . . . , Xn be a random sample of size n from a normal population with mean µX and variance σ ^2 . Let Y1, Y2, . . . , Ym be a random sample of size m from a normal population with mean µY and variance σ ^2 . Also, assume that these two random samples are independent. It is desired to test the following hypotheses H0 : σX = σY versus H1 : σX...