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

V. Composition: 20 pointsThe English test you are taking now probably reminds you of the other major English test you took for getting admission to a university for undergraduate study. While comparing between the experiences of taking these two tests, do you think your level of English proficiency has risen, stayed, or dropped during these years? What are the reasons for this change (or no change) to the proficiency level? Write a short paragraph about 120 words long to report your self-evaluation.

1. 依金融消費者保護法規定，期貨商應將交易人予以分類分級管理。試說明期貨商開戶徵信作業管理自律規則交易人開戶時期貨商應予交易人部位限制，及超過部位限制時應加收保證金之原則與加收標準。(10 分)

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)

(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 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%)