[試題] 100下 呂學一 機率 第二次期中考消失
課程名稱︰機率
課程性質︰必帶
課程教師︰呂學一
開課學院：電機資訊學院
開課系所︰資訊工程學系
考試日期（年月日）︰2012/5/7
考試時限（分鐘）：180
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
說明：總共八題，每題十五分，可按任何順序答題。歡迎使用事先準備的A4大抄。每題
難度不同，請審慎判斷恰當的解題順序。
第一題 What is the definition of Γ(r)? Prove that Γ(0.5) = √π.
第二題 A standard Cauchy random variable has density function
1
f(x)= ────── , where x can be any real number. Prove that if X is a
π‧(1+x^2)
standard Cauchy random variable, then so is 1/X.
第三題 The probability density function of X, the lifetime of a certain
type of electronic device (measured in hours), is given by
f(x)=｛ c/(x^2) if x>c
0 otherwise
where c is some positive constant.
(1) Find P(X>2‧c).
(2) What is the cumulative distribution function of X?
(3) If the lifetimes of distinct devices are independent, what is the
probability that, of 6 such types of devices, at least 3 will function for
at least 15 hours?
第四題 Let X and Y be two independent standard normal random variables.
Prove that X/Y has a Cauchy distribution.
第五題 Let X and Y be independent uniform distributions over [0,1].
(1) Find the joint density function of U = X and V = X-Y.
(2) Find the density function of V = X-Y.
第六題 The joint density function of X and Y is defined by
f(x,y)= ｛ x‧y if 0 < x < 1 , 0 < y < 2
0 otherwise
1. Are X and Y independent? Justify your answer.
2. Find the density function of X.
3. Find the density function of Y.
4. Find the joint distribution function of X and Y.
5. Find E[Y].
6. Find P(X + Y < 1).
第七題 Let X1, X2, X3 be independent and identically distributed continuous
random variables. Compute
1. P(X1 > X2 | X1 > X3)
2. P(X1 > X2 | X1 < X3)
3. P(X1 > X2 | X2 > X3)
4. P(X1 > X2 | X2 < X3)
第八題 Let X(1)≦X(2)≦…≦X(n) be the ordered values of n independent
uniform distributions over [0,1]. Prove that for any k with 1≦k≦n+1 and
any t with 0≦t≦1, we have
P( X(k) - X(k-1) ﹥t ) = (1-t)^n,
where X(0)=0 and X(n+1)= 1.
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