Please answer this question using R
Please answer this question using R 20. Let X1, X2, ..., X12 be a random sample...
Let X1, X2 ,, X8 be a random sample of size 8 from a POI(λ) distribution. The null hypothesis that λ = 0.25 will be rejected in favor of the alternative hypothesis that λ>0.25 if ∑X ≥5. i a. Find the probability of committing a type I error. (Hint: If Xi ~ POI(λ), then ∑Xi ~ POI(nλ)). b. Find the probability of committing a type II error if λ = 0.5 . c. Find the power of the test if...
1. Let X1, X2,... .Xn be a random sample of size n from a Bernoulli distribution for which p is the probability of success. We know the maximum likelihood estimator for p is p = 1 Σ_i Xi. ·Show that p is an unbiased estimator of p.
Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m =...
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,)...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
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(15 marks) 3. Xi, X2, Xs, X4 is a random sample from the Normal (0, 4) distribution. We want to test HO: θ 15 versus Hi: θ< 15. Let X_13x (a) Suppose we have two decision rules: Reject Ho if and only if () X <IS (II) X <12 Which one is better and why? (b) Instead of n 4, let n be an unknown. Let the decision rule now be: Reject Ho...
Let Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter p. Suppose that Y, X1 and X2 are independent. Proof using the de finition of distribution function that the the distribution function of Z =Y Xit(1-Y)X2 is F = pF14(1-p)F2 Don't use generatinq moment functions, characteristic functions) Xi and X2 independent random variables, with distribution functions F1, and F2, respectively Let Y a Bernoulli random variable with parameter...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
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).
6. Let Xi, X2, .., X6 be a random sample from a distribution with density function 820-1 for 0 < x 1 where θ > 0 f(x; 6) 0 otherwise The null hypothesis Ho : θ-1 is to be rejected in favor of the alternative Ha : θ 1 if and only if at least 5 of the sample observations are larger than 0.7. What is the significance level of the test