2. (10pts) Let X1, X2, , X20 be an i.i.d. sannple from a Normal distribution with mean μ and vari...
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).
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...
Suppose that X1, ..., Xn is a random sample from a normal distribution with mean μ and variance σ2. Two unbiased estimators of σ2 are 1?n 1 i=1 σˆ12 =S2 = n−1 Find the efficiency of σˆ12 relative to σˆ2. (Xi −X̄)2, and σˆ2= 2(X1 −X2)2
8. Let X, X2, , xn all be be distributed Normal(μ, σ2). Let X1, X2, , xn be mu- tually independent. a) Find the distribution of U-Σǐ! Xi for positive integer m < n b) Find the distribution of Z2 where Z = M Hint: Can the solution from problem #2 be applied here for specific values of a and b?
please answer with full soultion. with explantion. (4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the unknown parameter. Let X denote the sample mean. (Note that the mean and the variance of a normal N(μ, σ2) distribution is μ and σ2, respectively.) Is X2 an unbiased estimator for 112? Explain your answer. (Hint: Recall the fornula E(X2) (E(X)Var(X) and apply this formula for X - be careful on the...
2. (10pts) The following 16 random samples; 5.33, 4.25, 3.15, 3.70, 1.61, 6.40, 3.12, 6.59,3.53, 4.74, 0.11, 1.60, 5.49, 1.72, 4.15, 2.30, came from normal distribution with mean μ and variance σ2, i.e., Xi, X2' .. . , X16 ~ N(μ, σ*), with the density function (a) (4pts) Find the maximum likelihood estimates of μ and σ2, denoted with μ and σ2. (b) (4pts) Based on above μ and σ2, construct 95% confidence intervals for μ and σ2 separately. (c)...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be the...
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?
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση! (Xi-X)2 and S2-n-T Ση#1 (Xi-X)2 be the estimators of σ2 (i) Show that the MSE of s is smaller than the MSE of S2 (ii) Find E [VS2] and suggest an unbiased estimator of σ.