1. Suppose that X ∼ Normal(μ,σ2). NOTE: The second parameter here is the variance of X,...
1. Suppose that X ∼ Normal(μ,σ2). NOTE: The second parameter here is the variance of X, this is different to the book which uses the standard deviation as the second parameter. However, the more usual convention is to use the variance, and so that is what I will use in this class. Use the transformation technique to find and identify the distribution of (X − μ)/σ.
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1. Suppose that X ∼ Normal(μ,σ2). NOTE: The second parameter
here is the variance of X,...
Problem 1: confidence interval for a variance parameter for a normal distribution Let Ybe a normal...
Problem 1: confidence interval for a variance parameter for a normal distribution Let Ybe a normal random variable with mean μand variance σ2. Assume that μis known but σ2is unknown. Show that ((Y-μ)/σ)2is a pivotal quantity. Use this pivotal quantity to derive a 1-α confidence interval for σ2. (The answer should be left in terms of critical values for the appropriate distribution.)
(20 points) Suppose X-N(25, 81). That is, has a normal distribution with μ-25 and σ2-81. la....
(20 points) Suppose X-N(25, 81). That is, has a normal distribution with μ-25 and σ2-81. la. Find a transformation of X that will give it a mean of zero and a variance of one (ie., standardize X). lb. Find the probability that 18<x < 26. 1c. Supposing Y = 10 +5X, find the mean of Y 1d. Find the variance of Y
DISTRIBUTION OF SAMPLE VARIANCE: Xn ~ N(μ, σ2), where both μ and σ are Problem 4...
DISTRIBUTION OF SAMPLE VARIANCE:
Xn ~ N(μ, σ2), where both μ and σ are Problem 4 (25 points). Assume that Xi unknowin 1. Using the exact distribution of the sample variance (Topic 1), find the form of a (1-0) confidence interval for σ2 in terms of quantiles of a chi-square distribution. Note that this interval should not be symmetric about a point estimate of σ2. [10 points] 2. Use the above result to derive a rejection region for a level-o...
5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and...
5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and σ2 > 0. (a) Consider the statistic cS2, where c is a constant and S2 is the usual sample variance (denominator -n-1). Find the value of c that minimizes 2112 var(cS2 (b) Consider the normal subfamily where σ2-112, where μ > 0. Let S denote the sample standard deviation. Find a linear combination cl O2 , whose expectation is equal to μ. Find the...
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unk...
Please explain very carefully!
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unknown. (a) (5 marks) Let μ+σ~p denote the p-th quantile of the N(μ, σ*) distribution. What does this mean? (b) (10 marks) Determine a UMVU estimate of,1+ ơZp and justify your answer.
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 >...
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,...
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...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inea...
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...
x, and S1 are the sample mean and sample variance from a population with mean μ|...
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
Problem 4 - Bayesian inference with uniform prior The data are 21:n, the model is Normal(μ, σ*), with σ2 known. The problem is to obtain the posterior distribution of μ, p(p xỉ n, σ*)p(μ|xì n, σ2) wh...
Problem 4 - Bayesian inference with uniform prior The data are 21:n, the model is Normal(μ, σ*), with σ2 known. The problem is to obtain the posterior distribution of μ, p(p xỉ n, σ*)p(μ|xì n, σ2) when the prior po(A) is uniform in [-a, a] a. Using Bayes rule, obtain the expression of pĢi X1:n, σ*) as a function of a and the data. Be careful to handle all cases. Give and explicit simple expression for the normaliztion constant. You...
(4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the...
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...
(20 points) Suppose X-N(25, 81). That is, has a normal distribution with μ-25 and σ2-81. la. Find a transformation of X that will give it a mean of zero and a variance of one (ie., standardize X). lb. Find the probability that 18<x < 26. 1c. Supposing Y = 10 +5X, find the mean of Y 1d. Find the variance of Y
DISTRIBUTION OF SAMPLE VARIANCE:
Xn ~ N(μ, σ2), where both μ and σ are Problem 4 (25 points). Assume that Xi unknowin 1. Using the exact distribution of the sample variance (Topic 1), find the form of a (1-0) confidence interval for σ2 in terms of quantiles of a chi-square distribution. Note that this interval should not be symmetric about a point estimate of σ2. [10 points] 2. Use the above result to derive a rejection region for a level-o...
5.13. Suppose X1, X2, , xn are iid N(μ, σ2), where-oo < μ < 00 and σ2 > 0. (a) Consider the statistic cS2, where c is a constant and S2 is the usual sample variance (denominator -n-1). Find the value of c that minimizes 2112 var(cS2 (b) Consider the normal subfamily where σ2-112, where μ > 0. Let S denote the sample standard deviation. Find a linear combination cl O2 , whose expectation is equal to μ. Find the...
Please explain very carefully!
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unknown. (a) (5 marks) Let μ+σ~p denote the p-th quantile of the N(μ, σ*) distribution. What does this mean? (b) (10 marks) Determine a UMVU estimate of,1+ ơZp and justify your answer.
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 >...
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...
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...
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
Problem 4 - Bayesian inference with uniform prior The data are 21:n, the model is Normal(μ, σ*), with σ2 known. The problem is to obtain the posterior distribution of μ, p(p xỉ n, σ*)p(μ|xì n, σ2) when the prior po(A) is uniform in [-a, a] a. Using Bayes rule, obtain the expression of pĢi X1:n, σ*) as a function of a and the data. Be careful to handle all cases. Give and explicit simple expression for the normaliztion constant. You...
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...
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