108年 - 108 國立臺北大學_碩士班一般入學考試試題_金融與合作經營學系：統計學#146902

一、A finance PhD program recruits new students by two ways: Choosing the good-grade postgraduates (Inner Promotion, IP), or by opening exam competition (Open Competition, OC). P(IP) = 0.4, P(OC) = 0.6. After the first year qualification test, there are 90% IP and 75% OC students passed. If you randomly select a test-passed student, what is the probability the student recruited from IP? (10 %)

二、A mayor wants to know whether there exists seasonal differences about the visitors travelling to his city. He selects 1360 passengers and surveys their arrival time:

Please use Chi-square test under the significant level $\alpha=0.01$. (13%)

三、You participate a game by throwing a coin, which will be HEAD with P(H)=0.6 and TAIL with P(T)=0.4. The throwing results are independent. In the beginning, you own 5 dollars. If you throw a HEAD, you can get 1 dollar; however, if you throw a TAIL, you lose 1 dollar. This game will end when you own 15 dollars or lose all the money. What is the probability you own 15 dollars and lose all the money respectively? (12 %)

四、Two independent random variables X and Y. Y follows standard normal distribution, and X is uniform between [0,1]. Z = X + Y. Please derive the detail process that cov(X, Z) = var(X). (15%)

五、Suppose $X_1, X_2, \ldots, X_n$ are independent random variables and follow Poisson distribution with mean $\lambda$. Please use moment generating function to show that $\overline{X} = \sum X_i / n$ converges in probability to $\lambda$. (Hint: The distribution of $\overline{X}$ degenerates to a constant random variable.) (10%)

六、Suppose $X_1, X_2, \ldots, X_{10}$ are independent random variables and follow N(0, 1). Let $S = X_1+X_2+X_3+X_4+X_5$ and $W = X_6^2+X_7^2+X_8^2+X_9^2+X_{10}^2$. What is the probability $P(S/\sqrt{W} > 4.0324)$? (Rounded to third place.) (10%)

(a) When the sampling is done with replacement. (10%)

(b) When the sampling is done without replacement. (5%)

(a) \( \mathrm{P}(\mathrm{W}) \) , the probability of drawing a white chip. \( (10\%) \)

(b) \( \mathrm{P(B_1|W)} \), the conditional probability that bowl \( \mathrm{B}_1 \) had been selected, given that a white chip was drawn. (5%)