(b) \((5\%) f(x; \theta) = 1 / \theta, 0 < x < \theta < \infty\)

4. (10%) Assume that a stock is picked from a population of stocks at random. The average return of the population of stocks is 20%, and the standard deviation of the population of stocks is 2%. To calculate the maximum of the probability that a stock picked at random from this population, which has a return less than 16% or greater than 24%.

5. In a presidential election campaign, two potential candidates argue who has higher supporting ratio. They tend to incorporate into a candidate group, however, they have inconsistent ideas on who should be the vice-president candidate. Candidate A suggests if the poll demonstrates the difference of supporting rate between two candidate groups is not significant, he will support B to be the presidential candidate; otherwise, A should be the presidential candidate and B the vice. (a) \((7\%)\) If the poll surveys at \(95\%\) confidence level and the sampling error is approximately \(\pm 3\%\), how large of the sample size should be interviewed at least? \(\left\{Z_{0.975} = 1.96\right\}\)

(b) \((7\%)\) The results of the poll: Group (A,B) represents A the presidential candidate and B the vice; Group (B,A) vice versa.

| | Support rate (p) | Sample size (N) | | --- | --- | --- | | Group (A,B) | 48.3% | 2,046 | | Group (B,A) | 46.1% | 2,046 | How do they make the decision? (at 95% confidence level) 6. A bank constructs a computer system for reviewing small-size loans. The following data discloses the cases reviewed by the system: Total loan cases: 54,231 Passed cases by computer review: 49,082 Non-default cases after computer review: 46,880 Default cases after computer review: 2,202 Cases rejected by computer system: 5,149. Then the cases are sent to be reviewed by manually. Non-default cases after manual review: 4,585; default cases after manual review: 564. (a) (8%)By the perspective of the computer system, fill up the following confusion matrix with number of cases.

(b) (4%) What are the ratios of type I error and type II error?

(c) (14%) Calculate "Accuracy ratio", "Precision ratio", "Recall ratio", "False positive ratio", "False negative ratio", "F1-score".

7. (10%)Please derive when n→∞ and small p, binomial distribution will approach to Poisson distribution.

1. We have n activities \( a1, a2, \ldots, an \). Each activity \( ai \) has a start time \( si \) and a finish time \( fi \) where \( 0 \leq si < fi \). If \( ai \) is selected, activity \( ai \) takes place during the half-open time interval \( [si, fi] \). Activities \( ai \) and \( aj \) are compatible if the intervals \( [si, fi] \) and \( [sj, fj] \) do not overlap. In the Activity Selection Problem, we try to select a maximum-size subset of mutually compatible activities. Design an O(nlog n)-time algorithm to solve this problem. (15分)

2. Run the Prim's algorithm to find the minimum spanning tree for the following graph step by step. The starting vertex is the node a. (15分)

3. Run Radix sort on the following array: (329,457,657,839,436,720,355) step by step. (10分)

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