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題組內容

1. (16分) Let c be a constant and consider the density function for the random variable Y:

(d) Compute P(1 ≤ Y ≤ 1.5)

其他申論題

二、某船到港排水量11,000公噸,KG為6.7公尺。在第一艙裝貨2,000公噸,重心距龍骨6公尺,在中線右側 0.5公尺;第二艙裝貨800公噸,重心距龍骨6.5公尺,在中線左側3公尺;第三艙卸貨 1,600公噸,重心距龍骨7公尺,在中線左側1.2公尺。KM 為7.3公尺,求該船的傾斜角度?

(a) Find the value of c.

(b) Find the cumulative distribution function F(y).

(c) Compute F(1) and F(0.5)

2. (12分) Show that for 1 million flips of a balanced coin, the probability is at least 0.99 that the proportion of heads will fall between 0.495 and 0.505.

(a) Which of the four estimators are unbiased?

(b) Of the unbiased estimators, which has the smallest variance?

4. (12分) Commercial kerosene is stocked in a bulk tank at the beginning of each week.Because of limited supplies, the proportion X of the capacity of the tank available for sale and the proportion Y of the capacity of the tank actually sold during the week are continuous random variables. Their joint distribution is given byFind the covariance of X and Y.

5. (12分) Given the infinite population whose distribution is given by List the 16 possible samples of size 2 and use this list to construct the distribution of for random samples of size 2 from the given population.

6. (12分) A traffic engineer collects data on traffic flow at a busy intersection during the rush hour by recording the number of westbound cars that are waiting for a green light. The observations are made for each light change. Explain why this sampling technique will not lead to a random sample.

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