Let X1, X2, ..., X, denote a random sample from the normal distribution with mean u and variance o?. Which of ôi and ô2 are consistent estimators of o?.
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- I independent random variables X' and Y' have the, variances 36 and 16 respectively, find the correlation coefficient between (X+ Y) and (X- Y) assuming X and Y are zero mean.Suppose that f (x) = e=* for 0 < x Determine the mean and variance of the random variable.Suppose X, and X, have a bivariate normal distribution. Then, the condition distribution of X, given X, = x, follows a normal distribution with variance, o:(1-p)' of(1-p²) 0,(1-p²) a) b) c) d) 0,(1-p)
- let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x isLet Y, represent the ith normal population with unknown mean , and unknown variance of for i=1,2. Consider independent random samples, Y₁₁, Y2Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=-(Yu-Y)² (a) What is the distribution of Y,? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho: 4 = 40 versus Ha Pio when of is unknown and n, is small. (e) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a. (d) Define V₁, a function of S7, that has a chi-square distribution.Let X1,…,Xn be i.i.d. Exp(λ) random variables, where λ is unknown.What is the variance and quadratic risk of the unbiased estimator θ^ = n*mini(Xi).
- Derive the variance of the Student's t distribution using the definition Var (x) = ELX2] - EX)² .Let Y₂ represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁₁, Yi2,,Yin, of size ni, from the ith population with sample mean Y, and sample variance S?=²-1₁-1(Y - Y₁². (a) What is the distribution of Y;? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho: ₁ = o versus Ha Hiio when of is unknown and n; is small. (c) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.Let Y, represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S? = E(YY)². 7-1 (h) Find the standard error of U₂ in part (g), assuming that of = 0} = 0². (i) Discuss how the distribution of Y₁ - Y₂ can be used to test the equality of the two population means, ₁ and 2, when of = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ -Y2, for testing Ho: ₁ = H₂ against a two-sided alternative hypothesis at the a level of significance.
- X and Y be standard normal random variables with correlation p. Compute the joint and marginal distributions of X+Y and X-Y. Are X+Y and X−Y independent?Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya,Ya. Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=₁1(Yu - Y.². (g) For non-zero constants a,'s, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = o2 = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #, and p2, when of=o=o² is known. (j) Define appropriate rejection regions, in terms of Y₁ - ₂, for testing Ho: #₁ = ₂ against a two-sided alternative hypothesis at the a level of significance.Q2) Let X, X X. iid N(H,0), be a random sample from the normal population where u is assumed known. Please derive (a) The maximum likelihood estimator (MLE) for o'. (b)Is the MLE an efficient estimator foro? (c) Is the sample variance an efficient estimator for o ?