X,,X,...,X, random sample from U(0,0), 0>0. (1 – a)100% Lower C.I. for is
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- Consider 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 consistency8) Consider the data (1,1), (3,9), (4,16), If f[1,3] = 4, f[3,4] = z, f[1,3,4] 1. find z %3D2. Let X, X,.., x, be a random sample from a distribution with p.d.f. f(x;0)=0x, |>x>0 Is the MLE O EE of 0? AEE of 0?
- 1. Let y1,.. , Yn be a random sample from a uniform distribution on the interval (0, B). Let B1 2 Li=1 Yi and B, = ) Max(y1,.., Yn) be estimators for B. Find the sampling distribution of B2 Show that B, and B, are unbiased estimators for B. 11.3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.Let X, X2.**An be an i.i.d random sample from U(0,0), then the sufficient statistics for 0 is Select one: O a. h= max(x,) O b. h = min(x) C. no sufficient statistics for 0 d.
- 4. Let X,,X2, .. ,X„ be a random sample form a uniform distribution, f(X), = 0*(1– 0)*-X; 0 < x < 1. Assume T, = X, and T, = (HEx. Check the unbiasedness property for T, and T2. Find the MSE of T, and T2. Can you preferer one of them? Discuss your reasons. a. b. с.Please help meLet X1,X2.,X, and Y1,Y2,.Yn be two random sample of size n from a normal independent distributions Var(X)=1 and Var(Y)=20?. Let U = E-1(X; – X)² and W = E1(Y; – Y)? W a. What is the sampling distribution of the statistic U + 202 b. If the 90th percentile of the statistic (U + 202 is 33.20, what is the sample size n?
- 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<275. Let (*1, x2, .., rn) be n independent random sample of size n from a uniform distribution on the interval [01,02]; (a) find the c.d.f of, X(1), the minimum order statistic (b) find the c.d.f of, X(n), the maximum order statistic (c) Hence, find the mean and variance of X(1) and X(n)-a) Let X₁, X2, X3,..., X₁ be a random sample of size n from population X. Suppose that X follows an exponential distribution with parameter 0 and Y = ===1 X₁. n b) i) Show that y follows a chi-square distribution with 2n degrees of freedom. ii) What is the value of the sample size n, if P(Y > 20.4831) = 0.025? iii) What is the value of sample size n, if P(|Y - 2n| <40) ≥ 0.025? iv) Use the Central Limit Theorem to compute P(100 < Y < 115), for n = 40. Let X₁, X2, ..., X5 be a random sample of size 5 from the standard normally population. If the statistic Y is given by Y = X1 X2 X3 ₁²+X3² +X5² i) Briefly explain why Y follows a Student's t distribution with 3 degrees of freedom. ii) Compute the probability that Y is between 2.35 and 5.84. iii) What is the mean and variance of the statistic Y?