1. Let X have a Bernoulli distribution, where P(X 1-p and P(X 0 1-p. (a) For...
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:...
(b). Let x, X.,...,x. be a random sample from the Bernoulli (0) distribution 0). Find the most powerful test 1,:0= versus H,:6-of size a=0,011. (). What is the power of this test?
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (a) Find the maximum likelihood estimator (MLE) of θ (b) Give the form of the likelihood ratio test for Ho : θ-Bo versus H1: θ > θο. (c) Show that there is an appropriate statistic T - T(X) that has monotone likelihood ratio. (d) Derive the uniformly most powerful (UMP) level α test for versusS You must give an explicit expression...
The random variable X has two possible distributions: fo(x) = ze-z?/21(x > 0) or a2 /2 (a) Find the most powerful level α-0.05 test of Ho : X ~ 0(x) versus H1 : X ~ fı(x) on the basis of observing X only b) Calculate the power of your test in part (a). The random variable X has two possible distributions: fo(x) = ze-z?/21(x > 0) or a2 /2 (a) Find the most powerful level α-0.05 test of Ho :...
Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,)...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
You have observed one observation X from a distribution with probability density function fx (x) and support X = {x : 0 〈 x 〈 1} (a) Derive the most powerful α 0.05 test for testing Ho : fx(x) = 2x 1 (0 < x < 1) versus H1 : fx (x) = 5c4 1 (0 〈 x 〈 1). Be sure to give the rejection region explicitly. (b) Compute the power of the test You have observed one observation...
Let X be a sample of size 1 from a Lebesgue p.d.f. fo. Find a UMP test of size α (0, ) for Ho : θ--θ : θ-0, in the o versus H1 tollowing case: Let X be a sample of size 1 from a Lebesgue p.d.f. fo. Find a UMP test of size α (0, ) for Ho : θ--θ : θ-0, in the o versus H1 tollowing case:
Let Xi, .Χίο be a sample of iid Bin( 1, θ) random variables, and let e-{i : Σοί HA : θ 0.8. Determine a) the size of this critical region. b) the power of this critical region for 0 0.8. x,2 9} be a critical region for testing Ho:0 0.6 versus Let Xi, .Χίο be a sample of iid Bin( 1, θ) random variables, and let e-{i : Σοί HA : θ 0.8. Determine a) the size of this critical...