课程名称︰机率与统计 课程性质︰选修 课程教师︰黄乾纲 开课学院： 开课系所︰工程科学及海洋工程学研究所/系 考试日期（年月日）︰2012/11/8 考试时限（分钟）：180 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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