Homework Answers
We have, E(
)
= 
, i = 1(1)n.
X = 
Thus, E(X) = E()
i.e. E(X) = 
i.e. E(X) = 
i.e. E(X) = 
Hence, X will be an unbiased estimator of
if,
= 1.
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5. (Chihara and Hestelberg : Exercise 6.4.25) Let X1, X2, . . . , Xn be...
Let X1, X2, ..., Xn be a random sample of size n from a population that...
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.
Let X1, X2, ...,Xn be a random sample of size n from a population that can...
Let X1, X2, ...,Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) = max(X1, X2, ...,xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for 0.
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b)
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...
4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση!...
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.
explan the answer 1l. Suppose that X1, X2,... Xn are independent random variables. Assume that ElXi]...
explan the answer
1l. Suppose that X1, X2,... Xn are independent random variables. Assume that ElXi] /4 and Var(X )-σ, where i 1, 2, . .., n. If ai , aam. , an are constants. 1,a2, , an are constan (i) Write down expression for (i) E{Σ,i ai Xi) and (ii) Var(Li la(Xi). (i) Rewrite the expression if X,'s are not independent.
1. Let X1, . . . , Xn be a sample of size n from a...
1. Let X1, . . . , Xn be a sample of size n from a distribution with expectation μ (2X1 + X2 + . . . + Xn-1 + 2Xn)/(n+1)l be an estimator and variance σ . and let μ- for μ. Is it unbiased? asymptotically unbiased? consistent?
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It...
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically...
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
2. Let X1, X2,. ., Xn be a random sample from a uniform distribution on the...
2. Let X1, X2,. ., Xn be a random sample from a uniform distribution on the interval (0-1,0+1). . Find the method of moment estimator of θ. Is your estimator an unbiased estimator of θ? . Given the following n 5 observations of X, give a point estimate of θ: 6.61 7.70 6.98 8.36 7.26
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.
Let X1, X2, ...,Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) = max(X1, X2, ...,xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for 0.
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b)
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...
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.
explan the answer
1l. Suppose that X1, X2,... Xn are independent random variables. Assume that ElXi] /4 and Var(X )-σ, where i 1, 2, . .., n. If ai , aam. , an are constants. 1,a2, , an are constan (i) Write down expression for (i) E{Σ,i ai Xi) and (ii) Var(Li la(Xi). (i) Rewrite the expression if X,'s are not independent.
1. Let X1, . . . , Xn be a sample of size n from a distribution with expectation μ (2X1 + X2 + . . . + Xn-1 + 2Xn)/(n+1)l be an estimator and variance σ . and let μ- for μ. Is it unbiased? asymptotically unbiased? consistent?
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...
Question 6 Let X1, . . . , Xn denote a sequence of independent and identically distributed i.id. N(14x, σ2) random variables, and let Yı, . . . , Yrn denote an independent sequence of iid. Nụy, σ2) ran- dom variables. il Λί and Y is an unbiased estimator of μ for any value of λ in the unit interval, i.e. 0 < λ < 1. 2. Verify that the variance of this estimator is minimised when and determine the...
2. Let X1, X2,. ., Xn be a random sample from a uniform distribution on the interval (0-1,0+1). . Find the method of moment estimator of θ. Is your estimator an unbiased estimator of θ? . Given the following n 5 observations of X, give a point estimate of θ: 6.61 7.70 6.98 8.36 7.26
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