課程名稱︰統計與實習一
課程性質︰經濟系必修
課程教師︰陳旭昇
開課學院：社科院
開課系所︰經濟系
考試日期（年月日）︰2009/11/11
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Problem 1 (10%) Let X~U[0,1] and Y=-αlnX , α>0. Use the CDF technique to
find the distribution of Y .
Problem 2 (10%) Let{Xi}2 ~i.i.d N(μ,σ^2). Use MGF to determine the
i=1
distribtion of random variable Y=aX1+bX2+C , where a,b and c are constants.
Problem 3 (10%) George and John independently run a 100 meter dash to find out
who is faster. The chance that George finishes the race in 11 seconds is 75%,
while the chance that he finishes the race in 12 seconds is 25%. Similarly,
the probability that John finishes the race in 11 seconds is 80%,while the
probability that he finishes in 12 seconds is 20%. What is the probability
that their times are not tied?
Problem 4 (40%) The time needed (in minutes) to complete an exam is normally
distributed with mean 80 and variance 100.
1. What is the median time to complete the exam ?
2. What is the probability of completing the exam in one hour or less?
3. What is the probability that a student will complete the exam in more
than 60 minutes but less than 75 minutes?
4. Assume that the class has 130 students and that the exam period is 90
minutes in length. How many students do you expect will be unable to
complete the exam in the allotted time?
Problem 5 (30%) Let Y1 and Y2 have a joint density function given by
f(y1,y2)= k ,0≦y2≦y1≦1
0 ,otherwise.
1. Determine the constant k.
2. Find the marginal probability density functions of Y1 and Y2, and show
their support,respectively.
3. Are Y1 and Y2 independent?
4. Find P(Y1≦3/4∣Y2≦1/2).
5. Find the conditonal density function of Y1 given Y2,f(y1∣y2).
6. Find P(Y1≦3/4∣Y2=1/3).
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※ 編輯: lk806 來自: 140.112.243.69 (11/12 18:43)