課程名稱︰機率 課程性質︰系必修 課程教師︰林守德 開課學院：電資學院 開課系所︰資工系 考試日期（年月日）︰2010/4/1 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 : Probability 2010 Midterm1 (Prof. Shou-de Lin) 4/1 14:30~17:30pm Total Point:120 You can answer in either Chinese or English 1.What are the four essentials to define probability? Please illustrate them by an example (8pts) 2.(a)There are three boxes:a box containing two gold coins, a box with two silver coins, and a box with one of each. After choosing a box at random and withdrawing one coin at random, you find that the chosen coin is a gold coin. What is the probability that the remaining coin is also gold [5 pts] (b)For the same three boxes:After choosing a box at random, a person look at the box and then intentionally reveal one gold coin. Now what is the probability that the remaining coin is also gold[5 pts] 3.(a)Given a group of X people, what is the probability that at least two of them have the same birthday(assuming 365 days a year)[5 pts] (b)If you are told the probability in (a) is roughly 0.5, what is X? (hint:e^x is near to 1+x when x is small, and e^(-0.7) is near to 0.5)[8 pts] 4.Show that the variance of an exponential distribution(with mean θ) is θ^2 [8 pts] 5.Your company must make a sealed bid for a construction project. Your company will win if your bid is lower than other companies. If you win the bid, then you plan to pay another firm 100 thousand dollars to do the work. If you believe the minimum bid(in thousands of dollars) of other participating companies can be modeled as a uniform distribution in between(70,140), then how much should you bid to maximize your expected profit(10 pts)? 6.Let X be a random variable that represents the number of days that it takes a high-risk driver to have an accident. Assume that X has an exponential distribution. If P(X<50)=0.25, compute P(X>100|X>50). (5 pts) 7.Prof. Chou always finishes his lectures within 2 min when he hears the bell rings. Let X be a R.V. denotes the time that elapses between the bell-ring and the time he finish the class, and the pdf of X is kx^2, if 0<x<2, and 0 otherwise. This month Prof. Chou has 4 lectures, find the probability that within this minth Prof. Chou finishes hes class within 1 min only one time. (7 pts) 8.The pdf of X is f(x)=θx^(θ-1) ,0<x<1, 0<θ<∞, Let Y=-2θlnX, what is the pdf of Y? (5 pts) 9.Let X have a logistic distribution with p.d.f:f(X)=e^(-X)/(1+e^(-X))^2, -∞<X<∞ Let Y=1/(1+e^(-X)). What is the distribution of Y? (5 pts) 10.Your spaceship is observing the strength of a signal and made 10 observations as below. Please draw the CDF of this signal (10 pts)" 0.43, 0.74, 1.16, 3.48, 6.5, 0.68, 0.87, 2.08, 5.75, 13.9 11.The mgf of a r.v. X is 1/4e^t+3/4e^(3t) The mgf of a r.v. Y is 1/3e^t+2/3e^(2t), X and Y are independent. Q1:What is the mgf of W where W=X-Y?(8 points) 12.Captain Kirk's (Star Trek) spaceship is under the attack of some aliens from the "probability planet". All the space-computer functions are not working except the "random" function that returns a random number between [0,1]. To avoid crash, a sequence of data that follows certain distribution needs to be generated. As the chief scientist of this spaceship, can you try to rescue your crew by finishing the following task? (a)It is required to generate a sequence of 10000 data point that follows binominal distribution b(1000,0.3). Please describe how to achieve this task using only the random function? (6 pts) (b)It is required to generate a sequence of 1000 data that follows Poisson distribution of mean 3. How do you propose to do that using lnly the random functions? (10 pts) (Note: For this problem, you can either explain your approach in text or using pseudo code to write a program diong that.) 13.Throwing a needle (of length L) n times on top of a paper containing parallel lines of spacing D (assuming L<D), assume it is known that the number of times that the needle crosses a line is m, please prove that π=2 ×L ×n/(D ×m) [15 pts] --

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