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,X, be iid N(μχ, σ*), Yi, ,Yn be iid N(Pv, σ*), and X's and Question 2:...
Let X1, ...., Xm be iid N(μ1,σ2) and Y1, ..., Yn be iid N(μ2,σ2), and X's...
Let X1, ...., Xm be iid N(μ1,σ2) and Y1, ..., Yn be iid N(μ2,σ2), and X's and Y's are independent. Here -∞<μ1,μ2<∞ and 0<σ<∞ are unknown. Derive the MLE for (μ1,μ2,σ2). Is the MLE sufficient for (μ1,μ2,σ2)? Also derive the MLE for (μ1-μ2)/σ.
Assume X1, . . . , Xn iid normal with mean and variance ^2 , show...
Assume X1, . . . , Xn iid normal with mean
and variance
^2 , show that
a. X¯ and X^2 are independent.
b. Proof that X¯ is normally distributed with mean
and variance
^2/n.
c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ?
1) degrees of freedom.
d. Show that X¯ S/is
t distributed with (n ? 1) degrees of freedom
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This...
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and on 2, 6, and 2. Calculations are tedious here. Use the hints given in class and follow through
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
T (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) varia...
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for...
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for i = 1, 2, 3 (that is, each is normally distributed with mean mean E(Y) = i and variance V(X) = i2). For each of the following situations, use the Y, i = 1, 2, 3 to construct a statistic with the indicated distribution a) X2 with 3 degrees of freedom b) t distribution with 2 degrees of freedom c) F distribution with 1...
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(...
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown. Let ρ-r/of and g m/n, and consider the problem of unbiased estimation of u
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown....
3. Let X1, . . . , Xn be iid random variables with mean μ and...
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Could I grab some help on problem 2? Thank you 2. Suppose Yi, Yn are iid...
Could I grab some help on problem 2? Thank you
2. Suppose Yi, Yn are iid normal random variables with normal distribution with unknown mean and variance, μ and ơ2. Let Y ni Y. For this problem you may not assume that n is large. n (a) What is the distribution of Y? (b) What is the distribution of Z = (yo)' + ( μ)' + (⅓ュ)? (o) What is the distribution of ta yis (d) What is the distribution...
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed...
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
Assume X1, . . . , Xn iid normal with mean
and variance
^2 , show that
a. X¯ and X^2 are independent.
b. Proof that X¯ is normally distributed with mean
and variance
^2/n.
c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ?
1) degrees of freedom.
d. Show that X¯ S/is
t distributed with (n ? 1) degrees of freedom
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and on 2, 6, and 2. Calculations are tedious here. Use the hints given in class and follow through
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and...
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
t (0, c(X1-X2)2) įs a Let X, and X2 be iid. N(0, (Au)100% confidence interval for σ- 1) σ2) variables) . Find a constant so tha
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for i = 1, 2, 3 (that is, each is normally distributed with mean mean E(Y) = i and variance V(X) = i2). For each of the following situations, use the Y, i = 1, 2, 3 to construct a statistic with the indicated distribution a) X2 with 3 degrees of freedom b) t distribution with 2 degrees of freedom c) F distribution with 1...
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown. Let ρ-r/of and g m/n, and consider the problem of unbiased estimation of u
In 10. 11, Let X1, X2, . , Xn and Yi, Y2, . . . , Y,, be independent samples from N(μ, σ?) and N(μ, σ), respectively, where μ, σ. ơỈ are unknown....
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Could I grab some help on problem 2? Thank you
2. Suppose Yi, Yn are iid normal random variables with normal distribution with unknown mean and variance, μ and ơ2. Let Y ni Y. For this problem you may not assume that n is large. n (a) What is the distribution of Y? (b) What is the distribution of Z = (yo)' + ( μ)' + (⅓ュ)? (o) What is the distribution of ta yis (d) What is the distribution...
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
5 points Suppose that Yi N(0, σ ). Write out the likelihood for the data and show that it is equivalently to using ordinary least squares = β0 +너=12'ij8; + ei where ei, , en are iid. distributed from a
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