108年 - 108 國立臺北大學_日間學士班轉學生招生考試試題_金融與合作經營學日間學士班2年級：統計學#147627

科目：研究所、轉學考(插大)◆統計學 | 年份：108年 | 選擇題數：0 | 申論題數：9

試卷資訊

所屬科目：研究所、轉學考(插大)◆統計學

選擇題 (0)

申論題 (9)

一、(15%) If the moment-generating function of X is

$$M(t) = (2/5)e^t + (1/5)e^{2t} + (2/5)e^{3t},$$

find the mean, variance, probability density function of X.
二、(20%) Bowl B₁ contains 2 white chips, bowl B₂ contains 2 red chips, B₃ contains 2 white and 2 red chips, and B₄ contains 3 white chips and 1 red chip. The probability of selecting bowl B₁, B₂, B₃, B₄ are 1/2, 1/4, 1/8, and 1/8, respectively. A bowl is selected using these probabilities, and a chip is then drawn at random. Find

(a) \(\mathrm{P}(\mathrm{W})\), the probability of drawing a white chip.
(b) \(\mathrm{P(B_1|W)}\), the conditional probability that bowl \(\mathrm{B}_1\) had been selected, given that a white chip was drawn.
三、(20%) A bag contains 24 pieces of candy, of which 12 are peppermints and 12 are butterscotch. Let X equal the number of peppermints in a sample of five pieces of candy selected at random and without replacement from the bag. Find

(a) \(\mathrm{P}(X = 2)\)
(b) \(\mathrm{P}(X\leq 2)\)
(c) \(\mu = \operatorname {E}(X)\)
(d) \(\sigma^2 = \operatorname {Var}(X)\)
四、(20%) Let X₁, X₂, ..., Xₙ be a random sample from distributions with the following probability density functions:

$$f(x; \theta) = \frac{1}{2\theta^2} x^2 e^{-x/\theta}, 0 < x < \infty, 0 < \theta < \infty.$$

Please find the maximum likelihood estimator $\hat{\theta}$.
五、(25%) To test two method of instruction, 50 students are selected at random for each of two groups. At the end of the instruction period each student is assigned a grade (A, B, C, D, or F) by an evaluating team. The data are recorded as follows:

| | Grade | | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| | A | B | C | D | F | Total |
| Group I | 8 | 13 | 16 | 10 | 3 | 50 |
| Group II | 4 | 9 | 14 | 16 | 7 | 50 |

Please use a chi-square test with α = 0.05 to test the equality of the distributions of grade. The critical value of chi-square distribution for above test is as follows:

| r | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- |
| χ²₀.₀₅(r) | 5.991 | 7.815 | 9.488 | 11.070 | 12.590 |
| χ²₀.₀₂₅(r) | 7.378 | 9.348 | 11.140 | 12.830 | 14.450 |
| * P(≥χ²ₐ(r)) = α for X ~ χ²(r). | | | | | |