Let X₁,..., Xn be a random sample from a n(6,0²) population. Consider testing Ho: 01 ≤0 ≤ 02 versus H₁:0 <01 or 0 > 0₂. (a) Show that the test reject Ho if X>02+t-1.a/2√√/S2/n or X <01-tn-1.a/2√/S²/n is not a size a test. (b) Show that, for an appropriately chosen constant k, a size a test is given by reject Ho if X-|> k√√/S²/n,
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- Q Let Z be a standard normal random variable 2. Show that E[Z^*'] = n E [z"-¹]. to compute E [24] Use the resultConsider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θI got MLE = n/-∑logXi and UMVUE = (n-1)/∑logXi. Need help in proving consistencyLet X1,..., X10 be a random sample of size 10 from a N(u, o²) population. Suppose 10 Y =(X; - 4)? i=1 Find the probability that the random interval Y Y 20.5' 3.25 includes the point o?.
- der a random sample X1, X2, ..., X, having the pdf, f(r; 0) 1Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1Iwont helpPlease help me8. Let the random variable X have the pdf 2 x2 fx (x) = exp %3D - V2n 2 Find the mean and the variance of X. Hint: Compute E (X) directly and E (X²) by comparing the integral with the integral representing the variance of a random variable that is N(0,1). i DCO 04 < (X - 5)2 < 38.4).Let X1,X2,...,X25 be a random sample from N(u,36). Find UMPT of size a = 0.05 for testing HoH = 27 vs H,iH<27EL 466 416 13.) The continuous random variable (RV) X is uniform over [0,1). Given Y = -ln X what is P({0Let X₁,..., Xn be a random sample from a geometric distribution, X~ GEO(p). Here, -1 P[X = x] = p(1 − p)*−¹ for x = The method of moments unbiased estimator for Var [X] None of the other answers ·Σ" (X; – X)² i= X² n 1 [X² -x] = 1, 2,... n+1 = 1-p p² ist x, x, ... x be a random sample from an exponential distribution with parameter 0. Find sufficient estimator for '8'.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON