Homework Answers
What i have uploaded earlier that was also through gamma distribution since Chi-square is a Gamma(1/2,n/2) distribution. But this one is directly through gamma distribution and gamma moments. Kindly recheck it.
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5. Let X ~ Exp(A) with λ unknown, and suppose X1,X2 is a random sample of...
5. Let X ∼ Exp(λ) with λ unknown, and suppose X1, X2 is a random sample...
5. Let X ∼ Exp(λ) with λ unknown, and suppose X1, X2 is a random sample of size 2. Show that M = sqrt( X1 · X2 ) is a biased estimator of 1/λ and modify it to create an unbiased estimator. (Hint: During your journey, you’ll need the help of the gamma distribution, the gamma function, and the knowledge that Γ(1/2) = √ π.)
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows...
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution....
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
Let X1, X2,... X,n be a random sample of size n from a distribution with probability...
Let X1, X2,... X,n be a random sample of size n from a distribution with probability density function obtain the maximum likelihood estimator of λ, λ. Calculate an estimate using this maximum likelihood estimator when 1 0.10, r2 0.20, 0.30, x 0.70.
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
Let X1, X2, ...... Xn be a random sample of size n from EXP() distribution , ,...
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
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.
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF...
Let λ >0 and suppose that X1,X2,...,Xn be i.i.d. random variables with Xi∼Exp(λ). Find the PDF of X1+···+Xn. Use convolution formula and prove by induction
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
Let X1, X2,... X,n be a random sample of size n from a distribution with probability density function obtain the maximum likelihood estimator of λ, λ. Calculate an estimate using this maximum likelihood estimator when 1 0.10, r2 0.20, 0.30, x 0.70.
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
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.
Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...
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