



,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 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.
Suppose that X1, ..., Xm are iid Bernoulli(p), Y1, ...., Yn are iid Bernoulli(q), and that the X's are independent of the Y's where 0 < p < 1 is the unknown parameter with q = 1 - p. By means of the conditional distribution approach, show that sum of Xi - sum of Yi is sufficient for p. {Hint: Instead of looking at the data (X1, ..., Xm, Y1, ...., Yn), can one justify looking at (X1, ..., Xm,...
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...
Assume Y1, ...,Yn are IID normal random variables where mean μ and variance "2 are both unknown. Assume that ¯ Y = 0, and s, the sample standard deviation, equals sqrt(n). Compute a 1 − a confidence interval for the mean μ. Leave your answer in terms of ta/2, the critical value for a t distribution. How many degrees of freedom does this t distribution have?
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....
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...