91年 - 91 國立交通大學管碩士班考試入學試題_交通運輸研究所、運輸科技與管理學系：統計學#124768

(1) Evaluate P(X＞10) (7%)

(2) Calculate the mean and variance of the random variable X (10%)

2. Show that Y =has a chi-squared distribution with 1 degree of freedom when X has normal distribution with mean μ and variance σ². (10%)

3. Two friends plan to meet at the library during a given 1-hour period. Their
arrival times are independent and randomly distributed across the 1-hour
period. Each agrees to wait for 15 minutes, or until the end of the hour. If the
friend hasn't appeared during that time, the other will leave. What is the
probability that the two friends will meet? (15%)

4. A field traffic survey has shown that the number of vehicle arrivals per minute is Poisson distributed. Given that the average number of arrivals is 5 veh/min.
Determine the probability of a time headway (i.e., the time interval between
passages of consecutive vehicles moving in the same direction) greater than 1
minute? (6%)

(1) Find a 95% confidence interval for the actual proportion p of families in
the city who subscribe to the program. (5%)

(2) How large a sample is required if he wants to be at least 95% confident
that the estimated error of actual proportion p is within 0.02? (5%)

(1) Prove that the number of suits that Larry sells has a Poisson distribution
with mean λp. (10%)

(2) What is the probability that Larry does not sell any suits? (5%)

7. An equipment manufacturer has developed a new electrical product. He
claims that the product has a mean life length of 10,000 hours, with a standard
deviation of 150 hours. A consumer tested 16 random samples of the product
and obtained a mean life length of 9,775 hours. Evaluate at the 0.05 level of
significance whether the life length announcement for the new product is
reliable ? (12%)

(1) Assume that the error terms εᵢ are independent N(0, σ²) and that σ² is known. State the likelihood function for the n sample observations on Y and obtain the maximum likelihood estimator of β₁. Is it the same as the least squares estimator of β₁? (10%)

(2) Show that the maximum likelihood estimator of β₁ is unbiased. (5%)