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%)

2. 依期貨交易法第 63 條規定，期貨商之負責人及業務員為誇大、偏頗之宣傳或散播不實資訊等計有六種行為時，應負三年以下有期徒刑、科或併科新臺幣二百萬元以下罰金。試述法條所訂六種不得為行為之任何四種。(10 分)

3. 試簡述期貨服務事業之種類及其業務範圍如何。(10 分)

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

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

3. If X is a Poisson random variable with parameter λ ＞ 0, then theprobability 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, giveninformation 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%)