110年 - 110 國立臺灣大學_碩士班招生考試_統計碩士學位學程:統計學(G)#101826

科目：研究所、轉學考(插大)◆統計學 | 年份：110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)

1. (13%) State the definition of o -algebra. Explain the concept of σ -algebra.

2. (13%) Explain the concepts of probability measure and probability space.

3. (10%) Suppose we are playing draw poker. We are dealt (from a well shuffled deck) 5 cards which contain 4 spades and another card of a different suite. We decide to discard the card of a different suite and draw one card from the remaining cards to complete a flush in spades (all 5 cards spades). Determine the probability of completing the flush.

4. (10%) Let X have the pdf f(x) = 3x², 0 ＜ x ＜ 1, zero elsewhere. Consider a random rectangle whose sides are X and (I-X). Determine the expected value of the area of the rectangle.

5. (10%) If f(x) =,-1 ＜x ＜ 1, zero elsewhere, is the pof of the random variable X, find the pdfof Y =X².

6.(10%) If the random variable X has a Poisson distribution such that P(X = 1) = P(X= 2), lind P(X = 4).

7. (10%) Let Xand Y have a bivariate normal distribution with parameters , and ρ ＞ 0. If P(4 ＜Y ＜16|X = 5) = 0.954, determine ρ.

8. (10%) Let denote the mean of a random sample of size 128 from a gamma distribution with a = 2 and β = 4. ApproximateP(7＜＜9).

9. (14%) Let X₁,... , X_nbe iid n(H,σ²), where σ² is known. Show that the sample mean, T(X) = = (X₁ + .. + Xn)/n, is a sufficient statistic forμ.