Problem3 (15 points (a) (8 points) Let x, X, be a random sample from normal distribution NG, σ, ....
8) Let Yi, X, denote a random sample from a normal distribution with mean μ and variance σ , with known μ and unknown σ' . You are given that Σ(X-μ)2 is sufficient for σ a) Find El Σ(X-μ). |. Show all steps. Use the fact that: Var(Y)-E(P)-(BY)' i-1 b) Find the MVUE of σ.
and let (b) Let X, X,...,X, be a random sample form the normal distribution Nu,o) Σ- ΣΧ be the sample mean, S2 be the sample variance. j-1 n-1 Σ--Σ( - 1' -nΣΤ-β). (i) Prove that Using it, determine the distribution of X (ii) Find the m.g.f. of X. n ΣT- ) Σ- 7 7 n (iii) Indicate the distributions ofJ 2 , respectively. and (iii) Given that X and S are independent, derive the m.g.f of (n-15, and then, σ'...
Let X,, X,,...X be a random sample of size n from a normal distribution with parameters a. Derive the Cramer-Rao lower bound matrix for an unbiased estimator of the vector of parameters (μ, σ2). b. Using the Cramer-Rao lower bound prove that the sample mean X is the minimum variance unbiased estimator of u Is the maximum likelihood estimator of σ--σ-->|··( X,-X ) unbiased? c.
Let X,, X,,...X be a random sample of size n from a normal distribution with...
1. Let X,X, X, be a random sample from N(μ, σ*) and X and S2, respectively, be the sample mean and the sample variance. Let Xn+1 ~ N(μ, σ*), and assume that X,,X2,..XX+ are independent. Find the sampling distribution of [(X X) /n/(n
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and variance σ2 16. Find P (49 < Xs <51) and P (49< X <51) 2. Let Y = X1 + X2 + 15 be the sun! of a random sample of size 15 from the population whose + probability density function is given by 0 otherwise
1. Let Xi l be a random sample from a normal distribution with mean μ 50 and...
Let the random variable X follow a normal distribution with a mean of μ and a standard deviation of σ. Let X 1 be the mean of a sample of 36 observations randomly chosen from this population, and X 2 be the mean of a sample of 25 observations randomly chosen from the same population. a) How are X 1 and X 2 distributed? Write down the form of the density function and the corresponding parameters. b) Evaluate the statement:...
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Let X1, ..., Xn be a random sample from a Normal distribution with mean zero and standard deviation sigma. Let X bar and S^2 be the sample mean and sample variance, respectively. a. Find the constant c such that c(Xbar2) / S^2 has an F distribution. b. How many degrees of freedom are associated with this F distribution?
Let X1, ..., Xn be a random sample (i.i.d.) from a normal distribution with parameters µ, σ2 . (a) Find the maximum likelihood estimation of µ and σ 2 . (b) Compare your mle of µ and σ 2 with sample mean and sample variance. Are they the same?
Suppose x has a distribution with μ = 35 and σ = 18. (a) If random samples of size n = 16 are selected, can we say anything about the x distribution of sample means? Yes, the x distribution is normal with mean μ x = 35 and σ x = 4.5. No, the sample size is too small. Yes, the x distribution is normal with mean μ x = 35 and σ x = 18. Yes, the x distribution...