Homework 4(1).pdf

Homework 4Due on 3/30/20181.(Textbook Section 3.5-7,Page 91)Suppose that the joint p.d.f.of X and Y is asfollows:f(x,y)=2xey,for 0 x 1,and 0 y 0,otherwise.Are X and Y independent?2.Suppose that the joint p.d.f.of X and Y is as follows:f(x,y)=c(x+y2),for 0 x 1,and 0 y 10,otherwise.Determine(a)the conditional p.d.f.of X for every given value of Y,and(b)Pr(X 00,otherwise.Suppose also that for any given value X=x(x 0),the n random variables Y1,.,Ynare i.i.d.and the conditional p.d.f.g of each of them is as follows:g(y|x)=1x,for 0 y x0,otherwise.Determine(a)the marginal joint p.d.f.of Y1,.,Yn,and(b)the conditional p.d.f.ofX for any given values of Y1,.,Yn.4.Suppose that the p.d.f.of a random variable X is as follows:f(x)=12x,for 0 x 00,for x 0.Find the p.d.f.of Y=X1 X2.6.Let W denote the range of a random sample of n observations from a uniform distri-bution on the interval 0,1,Determine the value of Pr(W 0.9).2