P - 国立中山大学应用数学系 Department of Applied .doc

Ch.3 Homework Solution 3-5 (Buffon’s Neddle Problem) A needle of length L is dropped randomly on a plane ruled with parallel lines that are a distance D apart , where DL. Show that the probability that the needle comes to rest crossing a line is 2L/πD.Explain how this gives a mechanical means of estimating the value of π. Sol: Y :針中心點到底線之高 Y ~Uniform(0,D) Θ:針與X軸正向之夾角 θ ~Uniform(0,2π) ∴P(針與平行線相交)= 3-6 A point is chosen randomly in the interior of an ellipse:.Find the marginal densities of the x and y coordinates of the point . Sol: ∴ , 3-8 Let X and Y have the joint density (a) By integrating over the appropriate regions , find (i)P(XY),(ii)P(X+Y1),(iii)P(X?). Sol: (a)(i) (ii) (iii) 3-9 Suppose that (X,Y) is uniformly distributed over the region defined by 0y and -11 (a) Find the marginal densities of X and Y . (b) Find the two conditional density Sol: 0y and -11 , , ∴ (a) (b) 3-11 Let ,and be independent random variables uniform on [0,1].Find the probability that the roots of the quadratic are real . Sol: 3-12【Question】 Let (a)Find . (b)Find the marginal densities. 【Solution】 (a) (b) 3-15【Question】 Suppose that and have the joint density function (d)Find the marginal densities of and . Are and independent random variables? 【Solution】 (d) and are not independent. 3-17【Question】 Let be a random point chosen uniformly on the region (b)Find the marginal densities and using your sketch. Be careful of the range of integration. 【Solution】 (b) 3-18【Question】 Letandhave the joint densities function and 0 elsewhere. (b)Find . (c)Find the marginal densities of and . 【Solution】 (b) 3-19【Question】 Suppose that two components have independent exponentially distributed lifetimes, and , with parameters and , respectively. Find (a)and (b). 【Solution】 (a) (b) 3-23【Question】 Suppose that, conditional on , has a binomial distribution with trials and probability of success, and that is a binomial random variable with m trials