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題組內容

1 Let

be a random sample of size n. Using it, define the following; (18%)

(c) Sample mode

其他申論題

9. (10 points) Let P and Q be n x n matrices. We say P is similar to Q if there exists an invertible n x n matrix C such that C-lPC = Q. Prove that similar square matrices have the same eigenvalues with the same algebraic imilUplickies(代數重根數)•

【已刪除】10. (10 points) Let A be an n x n matrix such that Ax • Ay = x • y for all vectors x and y in . Show that A is an orthogonal matrix(正交矩陣).

(a) Sample mean

(b) Sample median

(d) Sample variance

(e) Sample standard deviation

(f) The coefficient of variation.

2. (a) What is meant by "Events At B are mutually exclusive'' ? (2%)

【已刪除】(b) Let be n mutually exclusive events such that for j =1,2,...,n，and whose union is the entire sample space S. For any given event B such that ，state the Bayes* Law Formula for the posterior probabilities (6%)

(i) Are A1 and A2 mutually exclusive? (2%)

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