89年 - 89 國立交通大學管碩士班考試入學試題_運輸工程與管理學系：統計學#124766

(1). The probability distribution of a random variable X. (5%)

(2). two events E₁ and E₂ are independent. (5%)

2. If a random variable X follows exponential distribution with parameter λ＞ 0,
then P(X ≤ x)=1-e^⁻λx, ∀x ＞ 0. Show that E{X} =  .(9%)

3. If X is a Poisson random variable with parameter λ ＞ 0, then the
probability P(x = k) =. Show that the mean of X is λ. (9%)

4. 某公司往返台北與新竹之長途巴士,其每班車每趟之載客人數的機率分配如
下表:

若單程票價為100元,且定義隨機變數為每班車每趟之票價收入,試問E(Y)與V(Y)各為何? (8%)

(1). Prove Bayes' Theorem:
The posterior probability P(Aᵢ|Bⱼ) of outcome Aᵢ occurring, given
information outcome Bⱼ occurs, is obtained as follows:
P(Aᵢ|Bⱼ) = , i = 1, 2,...
where P(Bⱼ|Aᵢ) is the conditional probability for the outcome Bⱼ occurring,
given outcome Aₖ (k = 1,2,...) occurs. (8%)

(2). Seventy-five percent of the graduates of a driver training school pass the official driver's test on the first attempt, and the other 25 percent fail. The school gives a pretest to graduates before they take the official test. Of the graduates who pass the official test on the first attempt, 90% passed the pretest. Of the graduates who fail the official test on the first attempt, 20% passed the pretest.
We wish to know the probability that a graduate can pass the official test on the first attempt, given that he passed the school pretest. (6%)

6. 交通大學現有8000位學生,根據過去的一項調查,發現這些學生每月平均零用錢為8200元,標準差為1200元。則欲進行抽樣調查,要求誤差在 95% 的信賴水準下,不超過加減5%的時候,至少應抽選多少樣本數?(15%)

7. 中壢客運與新竹客運同時都行駛中壢/新竹線,剛考上交通大學的王同學想知道哪一家較受乘客歡迎,於是他於不同時段抽查兩家公司的每車載客數,
結果如下:

若假設每車載客人數為常態分配,試問兩家公司之每車載客數是否相等?(15%)

8.剛退伍的小智想加盟漫畫王連鎖店,來請教你的意見。連鎖店的企畫師告訴小智連鎖店的收益和周圍一公里以內的競爭店的多寡、家庭的平均所得、出租書的價格有關,但是出租書的價格幾乎相同,因此可以不考慮,企畫師提供的其他資料如下:

試問:迴歸方程式為何?模式是否可被接受?自變數的聯合解釋能力如何?(20%)