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

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.

(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.

(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:

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: