113年 - 113 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147065

1. A hospital obtains 50% of its flu vaccine from Company A, 40% from Company B, and 10% from Company C. From past experience, it is known that 4% of the vials from A are ineffective, 3% from B are ineffective, and 6% from C are ineffective. The hospital tests five vials from each shipment. If at least one of the five is ineffective, find the conditional probability of that shipment's having come from B. (10%)

2. Tiffany and John decide to meet at the train station. If each person independently arrives at a time uniformly distributed between 1 p.m. to 1:30 p.m., calculate the probability that the first to arrive has to wait longer than 10 minutes. (10%)

(1) Find the p.d.f. of the sample median \( Y_{2} \). (5%)

(2) Compute the probability that \( Y_{3} \) is less than two. (5%)

(3) Determine \( P(1 < Y_1) \). (5%)

4. A random sample $X_1, X_2, \ldots, X_n$ of size n is taken from a Poisson distribution with a mean of $\lambda, 0 < \lambda < \infty$. Show that $\bar{X}$ is the minimum-variance unbiased estimator for $\lambda$. (15%)

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