课程名称︰机率导论 课程性质︰数学系大二必修 课程教师︰郑明燕 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2014/6/19 考试时限（分钟）：13：20～15：20 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. Suppose that random vector (X,Y) has a joint probability density function (pdf) given by ︴6(y－x) , if 0 < x < y < 1, f(x,y) = ︴ ︴0 , otherwise (a) (6 pts.) Verify that f(x,y) is a valid pdf. (b) (9 pts.) Find the conditional pdf of X|Y = y for any 0 < y < 1. (c) (10 pts.) Compute Var(X), Var(Y) and Cov(X,Y). (d) (5 pts.) What is the best constant guess of X under the mean squared * error criterion? That is, find the constant a that minimizes 2 E[(X － a) ] over all a∈R. (e) (5 pts.) What is the best linear prediction of X based on Y under the _ _ mean squared error criterion? That is, find a + bY that minimizes 2 E[(X － a － bY) ] over a,b∈R. (f) (5 pts.) What is the best functional prediction of X based on Y under * the mean squared error criterion? That is, find g (Y) that minimizes 2 E[(X － g(Y)) ] over functions g on R. 2. (10 pts.) In a class with 15 boys and 10 girls. boys have probability 0.6 of knowing the answer and girls have probability 0.7 of knowing the answer to a typical question the teacher asks. Assume that whether or not the students know the answer are independent events. Find the mean and variance of the number of students who know the answer. 3. The probability mass function of a Poisson(λ) distribution is given by －λ k e λ f(k) = —————, k = 0,1,... k! (a) (10 pts.) Show that the moment generation function of a Poisson(λ) distribution, λ > 0, is given by t M(t) = exp[λ(e －1)]. _ 1 n (b) (10 pts.) Find the probability distribution of X = —Σi=1 X , if X , n n i 1 ..., X are i.i.d Poisson(λ) random variables. n 4. (a) (10 pts.) State the Weak Law of Large Numbers for i.i.d samples. (b) (10 pts.) State the Central Limit Theorem for i.i.d samples. 5. (10 pts.) Suppose that X is a random variable following the Poisson(225) distribution; i.e. -225 k e 225 P(X = k) = ——————, k = 0,1,2,... k! Find an approximate value of the probability P(200 ≦ X ≦ 250). --

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