The random variable X has two possible distributions: fo(x) = ze-z?/21(x > 0) or a2 /2 (a) Find t...
1. Let X have a Bernoulli distribution, where P(X 1-p and P(X 0 1-p. (a) For a random sample of size n = 10. test Ho : p $ versus H1 : p > 흘. Use 10 the critical region {ΣΧί 6) i. Find the power function, and sketch it. ii. What is the size of this test? (b) For a random sample of size n = 10: i. Find the most powerful test of Ho : p = 흘...
~ fo(x) On the basis of observing , find the most powerful size versus Hi: X~ fi(x), where fo(x) is the standard normal pdf and 0.05 test of H0 : that is fi(x) is the standard LaPlace pdf. Can you redo this problem when you have observed X1, X2 iid?
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:...
Suppose X ~ Íx (x10) (82+1-2)|(0 < x < 1), where 0 < θ 2, Find the UMP level α 0.02 test of Ho : θ 〉 1 versus H1 : θ 〈 1 on the basis of observing X. Determine the numerica 1 critical value for the test.
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...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
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:
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
Mean = -0.92809
This file contains independent observations on a random variable X. It is known that a2-1.4246. a. Test Ho : E(X) 1.6 against H1 : E(X)>1.6 using a significance level of 1%. In your answer, you should state the null and alternative hypotheses, the significance level, the PDF of the test statistic under the null hypothesis, the value of the test statistic, and your conclusion. b. Do you need to make any assumptions in order for your test...
Problem 2: Consider two probability density function on [0,1] : fo(x) = 1 and fi(x) = 2x (a) Construct the most powerful test for Ho : X~ fo against H:Xfi with the signifi- cance level x = 0.1 (b) Find its power. (c) Suppose we observe X= 5/6. What is the p-value of the test in 2(a).