1 Find the mean vector, the covariance matrix, and the correlation matrix of the random vector (X Y )T.
1 A random vector X = (X1 X2 X3 X4)T has mean vector and covariance matrix given as
1 Let and . Please findE(X1)E(AX1)cov(X1)var(AX1)E(X2)E(BX2)cov(X2)cov(BX2)
1 Repeat Exercise 2, but with A and B replaced by
1 Let X = (X1 X2 X3)T be a random vector with X ∼ N(μ, Σ) with
1 Which of the following random variables are independent? Please explain.X1 and X2X2 and X3X1 and X3(X1, X2) and X3 and X3
1 Consider the random vector X in Exercise 2.Find the distribution of X2 + X3 + X4.Find the distribution of 3X2 − 2X3 + X4.Find the joint distribution of X2 + X3 + X4 and 3X2 − 2X3 + X4.Find the distribution of X1 − X2 + 2X3 + X4Find a 2×1 vector c such that X2 and are independent.Find a 2×1 vector c such that X2 and are independent.
2 Consider the random vector in Exercise 4.Find the conditional distribution of X1, given that X3 = x3.Find the conditional distribution of X1, given that X2 = x2.
3 Consider the random vector X in Exercise 2.Find the conditional distribution of X1, given that X2 = x2 and X3 = x3.Find the conditional distribution of X2, given that X3 = x3 and X4 = x4.Find the conditional distribution of X3, given that X2 = x2 and X4 = x4.Find the conditional distribution of (X2 X3)T, given that X4 = x4.
4 Calculate by hand the maximum likelihood estimates of the mean vector μ and the covariance matrix Σ of (X2 X3)T based on the first five observations of the last two variables in Table 2.1, assuming the observations are from a bivariate normal population.
5 Consider a random sample of size n = 3 from a bivariate normal population as shown in the following table.
| x 1 | x 2 |
|---|---|
| 5 | 8 |
| 9 | 5 |
| 7 | 2 |
1 Evaluate the T2-statistic used to test H0 : μ = μ0 based on this data set, where μ0 (8 4)T. What is the distribution of the T2-statistic in this case?
1 Consider the data from a bivariate normal population in the following table:
| x 1 | x 2 |
|---|---|
| 4 | 14 |
| 10 | 11 |
| 8 | 11 |
| 10 | 12 |
1 Evaluate the T2-statistic for testing H0 : μ = (9 13)T using the data.Specify the distribution of the T2-statistic from (a).Using (a) and (b), test H0 at the α = 0.05 level. What conclusion do you reach?
1 Use the data in Exercise 10 to calculate the likelihood ratio test statistic LR using (3.23). Verify the correctness of (3.24) for this data.
2 Consider the hot rolling process as described in Example 3.2. Check if the mean side temperatures for the defective billets at the following locations along Stand 5 deviate significantly from the nominal values:locations 10 and 15 with nominal mean temperatures equal to 1852.6 and 1872.4, respectively.locations 6, 7, and 8 with nominal mean temperatures equal to 1878.0, 1868.5, and 1860.6, respectively.locations 17, 18, 19, and 20 with nominal mean temperatures equal to 1876.7, 1875.7, 1872.7, and 1868.5, respectively.
3 Perform Bayesian inference for the mean of side temperatures at locations 6, 7, and 8 based on the data set side_temp_defect. Please use the sample covariance of all the data at these three locations as the true covariance matrix and assume it is known. The mean and covariance matrix of the prior distribution is:
1 Please find the posterior distribution of the mean temperatures at locations 6, 7, and 8 based on the first five (n = 5) observations, and the posterior distribution based on the first 100 (n = 100) observations, respectively. Comment on how the posterior distributions are different for different sample sizes. And compare the MAP estimate with the MLE.
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