99年 - 99 國立交通大學管碩士班考試入學試題_運輸科技與管理學系_運管系乙組一般生：統計學#123996

(a) Find the value of c.

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

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

(d) Compute P(1 ≤ Y ≤ 1.5)

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 by

Find 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.

(a) Show thatis an unbiased estimate of the binomial parameter p.

(b) Show that is not an unbiased estimate of the binomial parameter p.

8. (12分) Let Y = 5 - 3X. Show that the correlation coefficient ρ{X,Y} =, is equal to -1.0.