課程名稱︰機率與統計 課程性質︰選修 課程教師︰黃乾綱 開課學院： 開課系所︰工程科學及海洋工程學研究所/系 考試日期（年月日）︰2012/11/8 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 : 1.[CH1](20%)Consider a binary code with 5 bits(0 or 1) in each code word. An example of a code word is 01010. Ineach code word, a bit is a zero with probability 0.8, independent of any other bit. a. How many different code words are there? b. How many code words have exactly three 0's? c. What is the probability if the code word 00111? d. What is the probability that a code word contains exactly three ones? 2.[CH2](15%)The random variable X has CDF 0 0<-3 Fx(x)= 0.4 -3<=x<5 0.8 5<=x<7 1 x<=7 a. Write Px(x), the PMF of X. Find the expected value of random variable X. b. Let W=g(X)=-X. Find Pw(w) and Fw(w) and E[W]. c. Find PX|B(x) where the condition B = {X>0} what are E[X|B] and VAR[X|B]? 3.[CH2](10%)The binominal random variable X have PMF, Px(x) = (C5取x)(1/2)^5 a. Find the standard deviation of the random variable X. b. What id P[ux-sigmax<X<ux+sigmax], the probability that X is within one standard deviation of the expected value? 4.[CH2](5%)Prove the geometric(p) random variable X has expected value E[X]=1/p 5.[CH3](10%) Random variable Y has CDF 0 y<-1 FY(y)= (y+1)/2 -1<=y<1 1 y>=1 a. What is E[Y]? b. What is Var[Y]? 6.[CH3](10%) Y is an exponential random variable with variance Var[Y]=25 a. What is the PDF of Y b. What is E[Y^2] 7.[CH3](10%) The peak temperature T, as measured in degrees Fahrenheit, on a July day in New Jersey is the Gaussian (85, 10) random variable. a. What is P[T<60]? b. What is P[70<T<100]? 8.[CH3](5%) X is a Gaussian random variable with E[X]=0 and P[|X|<= 10]=0.1, What is the standard deviation sigmax? 9.[CH3](15%) X is a uniform random variable with parameters -5 and 5. Given the event B={|X|<=3} a. Find the conditional PDF, fX|B(x) b. Find the conditional expected value, E[X|B] c. Find the conditional variance, Var[X|B] -- --

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