b) Let X to be a random sample from a distribution with a probability density function given by (ex-1,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.
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- Suppose that X is a random variable having the following probability density function given by (2(0-x) x>0 f(x) = elsewhere Find the value of c such that an interval from x to cx is a (1-a)100% confidence interval for the parameter 8.Let Y1, Y2,..., Y, denote a random sample from the density function given by 1 yª-'e=y/®, y> 0, f(y[a, 0) = elsewhere, where a > 0 is known. a Find the MLE Ô of 0. b Find the expected value and variance of ê. c. Is the MLE ô an unbiased estimator for 0?b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.
- With reference to Definition 4, show that μ0 = 1and that μ1 = 0 for any random variable for whichE(X) exists.a) Suppose that X is a random variable having the following probability density function given by (2(0-x) 02 f(x) = -, x > 0 (0, elsewhere Find the value of c such that an interval from x to cx is a (1- a)100% confidence interval for the parameter 0.b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 < x < 1, 0 € {1,2}, elsewhere O It is desired to test a null hypothesis Ho: 0 = 1 against the alternative hypothesis H₁:0 = 2. Suppose that the test rejects Ho if x ≥ ii) Calculate the power of the test.
- Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?b) The probability density function of random variable X is given as: - Вх {*5 if 0 < x < 1 otherwise f(x) = If the expected value of the random variable X is 1/3. Find the values A and B.6. Suppose that the random variables X and Y have joint probability density function given by x+y, 0Please answer number 2b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.