課程名稱︰機率與統計 課程性質︰系訂必修 課程教師︰謝宏昀 周俊廷 張時中 葉丙成 開課學院：電機資訊學院 開課系所︰電機工程學系 考試日期（年月日）︰2010/6/24 考試時限（分鐘）：180分鐘 是否需發放獎勵金：是 謝謝 （如未明確表示，則不予發放） 試題 : Probability and Statistics (Spring 2010) Final Exam (1) (10％) Consider the problem of coordinate system transformation in the 2- dimensional plane. A point (Ｘ,Ｙ) in Cartesian coordinates can be transformed into polar coordinates (Ｒ,Θ) such that Ｘ＝ＲcosΘ Ｙ＝ＲsinΘ, where Ｒ≧0 and Θ€[0,2π]. Let Ｘ and Ｙ be independent zero-mean, unit- variance Gaussian random variables. (a)(6％) Find the joint probability distribution function (PDF) f(r,θ) of random variables Ｒ and Θ. Are random variables Ｒ and Θ independent? (b)(4％) Find the PDF of derived random variable Ｒ^2. (2) (10％) Let X be an n-dimensional random vector with joint PDF f_X(x) = exp[-(1/2)x'(Cx^-1)x] / (2π)^(n/2)[det(Cx)]^(1/2) where the invertible matrix Cx is the covariance of X. (a)(5％) Show that for any nonzero vector x,-(1/2)x'(Cx^-1)x < 0 (Note: no equality sign.) (b)(5％) Show that X has independent components if and only if Cx is a diagonal matrix. (3) (10％) The Ｑ function for the standard normal distribution is frequently used in many applications. However, since it does not have a closed form, many bounds have been developed for approximating its value. (a)(5％) Use the Cgernoff bound to show that Ｑ(z)≦e^[-(z^2)/2]. (b)(5％) The Chernoff bound is in fact not very tight for the Ｑ function, especially when z is large. Can you provide a tighter bound? Alternativ- ely, you can try to prove the following bound for z>0: Ｑ(z) < [1/(z√2π)]e^[-(z^2)/2]. (4) (10％) The theorm of iterated expectation E[E[X|Y]] = E[X] states that the expectation of a random variable X can be obtained through its conditional expectation given another random variable Y. Similarly, the variance of a random variable can be obtained through its conditional variance. (a)(5％) Show that Var[X] = E[Var[X|Y]] + Var[E[X|Y]]. (b)(5％) Let {X1,X2,...} be a collection of idd random variables, and let N be a nonnegative integer-valued random variable that is independent of {X1,X2,...}. If R=X1+X2+...+XN, find the variance of R in terms of the exceptations and variances of X and N. (5) (10％) The Law of Large Numbers is sometimes known as the first fundamental theorm of probability due to its importance in probability and statics. (a)(6％) Let X1,X2,... be a sequence of idd random variables with expectat- ion μ and variance σ^2. Let Mn = (X1+X2+...+Xn)/n be the sample mean. What do the Central Limit Theorem and the (Weak) Law of Large Numbers state about the random variable Mn as n→∞? Are these two theorems con- sistent eith each other? (b)(4％) The relative frequency of an event is the proportion of the time that the event is observed in a large number of runs of experiments. Explain how the Law of Large Numbers can be used to show that the relat- ive frequency of an event A converges to its actual probability P[A] as the number of runs n→∞. (6) (10％) Let X = [X1 X2 X3]' be a 3-dimensional Gaussian random vector with μx = [2 4 6]' and covariance ┌ 8 -4 2 ┐ Cx = │ -4 8 -4 │ └ 2 -4 8 ┘. (a)(5％) Calculate the correlation matrix Rx. (b)(5％) Write down the joint PDF of the first two components of X, i.e. f_X1,X2(x1,x2). (7) (14％) Consider a random variable X eith the following PDF f_X(x) = (1/6)exp[-x/2]u(x) + [1/(3√(2π)]exp[-(x-5)^2/8], where u(x) denotes the unit-step function. (a)(8％) Derive the moment feneration function of X. Detailed derivation has to be given to get full credit. (b)(6％) Compute μx and (σx)^2 (8) (20％) Consider the restuerant named "It is so good to be rich". The number of customers visiting the restaurant during Monday,N,is Poisson distributed with μN = 50. The chance of each customer to order fried noodle is 0.7,and the orders of different customers are assumed to be independent. (a)(8％) The owner of the restaurant has to determined how many orders of fried noodle that he has to prepared in advance. Let X denote the number of fried noodle ordered on Monday. Please derive the PMF of X. What kind of distribution is this? (b)(6％) If the time needed for the owner to cook each order of fried noodle is idd with exponential distribution of mean 5. Let T denote the total amount of time the owner speeds on cooking fried noodle during the whole Monday. What is the moment generating function of T? (c)(6％) Compute μT and (σT)^2 (9) (6％) As you may or may not know that in the adventure of Midterm 2009, Dr. Fones got fooled by a dishonest mummy. Dr.Jones was so mad about it and just could not let go even after a year has passed. He got so mad that he decided to do a research about "What is the chance of meeting a dishonest mummy?" He visited Egypt again to conduct the research. Inside the pyramid, he found a lot of mummyies dancing in a secrete chamber. Being so mad and forgot his fear for mummies, Dr. Jones showed an apple to each mummy he met and asked the mummy what he saw. Out of the 1000 mummies he met, 650 of them answered "banana", 300 of them answered "apple", and 50 of them just murmured with unrecognizable sounds. From the experiment, Dr. Jones conclu- ded in his research paper with the following statement: "The chance of meeting a lying mummy is 65% with an accuracy of ±5%." You are an editor of a journal which only publishes research papers with results of high confidence level (i.e. 1-α≧90%). Please compute the conf- idence coefficient 1-α corresponding to the conclusion of Dr. Jones to determine whether you shoule reject his paper. --

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