課程名稱︰機率導論
課程性質︰系定必修
課程教師︰彭(木百)堅
開課系所︰數學系
考試時間︰2005.1.11
試題：
The examination begins at 15.30 and ends at 17.20. All working is to be
shown. If you use theorems or results which have not been taught, you
must give the proofs.
1.Let X be a continuous random variable with probability distribution
function (pdf):
f(x) =┌ cx exp(-x^2 / 2), if x≧0
└ 0 , if x< 0.
Find (a) c; (b) E[X]; (c) Var(X). [12 marks]
2.For a group of 100 people, calculate:
(a) the expected number of days of the year that are birthdays of exactly
3 people;
(b) the expected number of distinct birthdays. [11 marks]
3.A coin having probability p of coming up heads is continually flipped
until both heads and tails have appeared. Find
(a) the expected number of flips;
(b) the probability that the last flip lands heads. [11 marks]
4.The lifetime of a light bulb is an exponential random variable with a
mean of 100 hours. A case contains 60 independent light bulbs. Let γ
denote the probability that the total life time of the 60 light bulbs is
more than 7000 hours.
(a) Find an upper bound for γ using the Markov inequality;
(b) Approximate γ using the Cental Limit Theorem. [10 marks]
5.Let X be a standard normal random variable, I a random variable
independent of X with P{I=1} = P{I=0} = 1/2 and let Y be as defined below:
Y = X when I = 1, Y = -X when I = 0
(a) Find the probability density function (pdf) of Y;
(b) Show that Cov(X, Y) = 0;
(c) Are X and Y independent? Justify your answer. [12 marks]
6.Let X and Y be independent standard normal random variables. Find the
probability density function (pdf) of Z = X/Y. [14 marks]
7.Let X and Y be independent exponential random variables with the same
parameter λ.
(a) Find the joint probability density function of U = max(X, Y) and
V = min(X, Y).
(b) Are U and V independent? Justify your answer. [15 marks]
8.Let X be a binomial random variable with parameters (n, p). Suppose p is
also a random variable Y with uniform distribution on (0, 1). Find
(a) E[X]; (b) Var(X); (c) the moment generating function of X.
[15 marks]
教授提供的中譯試題：
考試時間為15.30至17.20。每一題接應詳述過程，理由。如引用未學過的定理或結果，
應先予證明。
1.設 X為具有以下的機率分佈(pdf) 的連續隨機變數：
f(x) =┌ cx exp(-x^2 / 2), if x≧0
└ 0 , if x< 0
試求(a) c； (b) E[X]； (c) Var(X)。 [12分]
2.在一百個人當中，試求： (註：一年以 365天計)
(a) E[X], 當 X為 S的數量，其中 S為剛好有三個人的生日落在此日的日子；
(b) E[Y], 當 Y為一百個人當中出現不同生日的數量。 [11分]
3.如果擲某一個銅板出現正面的機率為 p。我們一次連一次的擲此銅板一直到正面與反
面都出現過。令 X為所需要擲銅板的次數。試求
(a) E[X]；
(b) 最後一次擲銅板出現正面的機率。 [11分]
4.燈泡的生命是以 100小時為平均的指數隨機變數。令γ為60個獨立燈泡總共生命超過
7000小時的機率。
(a) 用 Markov 不等試求γ的上界；
(b) 用 Central Limit Theorem 求γ的近似值。 [10分]
5.設 X為標準常態隨機變數， I為與 X獨立的隨機變數，P{I=1} = P{I=0} = 1/2，
Y定義如下：
Y = X 當 I = 1， Y = -X 當 I = 0。
(a) 試求 Y的機率分佈(pdf) ；
(b) 試證 Cov(X, Y) = 0；
(c) X與 Y為獨立嗎？試說明。 [12分]
6.設 X與 Y為標準常態分佈之兩獨立隨機變數。試求 Z = X/Y 的分佈(pdf) 。
[14分]
7.設 X與 Y是以λ為參數的指數分佈獨立隨機變數。
(a) 試求 U = max(X, Y) 與 V = min(X, Y) 的聯合機率分佈(joint pdf)；
(b) U與 V為獨立嗎？試說明。 [15分]
8.令 X是以(n, p)為參數的二項式隨機變數。設 p亦為隨機變數，其分佈為(0, 1)上的
均勻分佈。試求：
(a) E[X]； (b) Var(X)； (c) X 的 moment generating function。[15分]
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