1. Given the discrete uniform population f(x)= 1/3, x= 2,4,6. Find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means to be measured to the nearest tenth.
2. Let Y1, Y2, Y3,........., Yn be iid (independent and identically distributed) Expo(1). Show that n! = (2pi)^(1/2) (n)^(n+n/2) (e)^(-n), n>0. (*Hint: Use the Central Limit Theorem)
3. Let (X,Y) be a random point over the unit circle R, R = {(x,y)| x^2 + y^2 <= 1}. Find the expected value E[(x^2 + y^2)^1/2].
4. Let X and Y be iid Expo(1).
(a) Find the densities of U=X+Y and V=X /(X+Y)
(b) Show that U and V are independent.
(c) Find E(U), E(U^2), Var(U), E(V), E(V^2), and Var(V).
5. A random sample of size 5 is drawn from the pdf fy(y)= 2y, 0<=y<=1. Calculate P[Y(1)<0.6<Y(5)]. (Note: "1" and "5" are underscript, order statistics question)
更新1:
Correction in #2: Show that n! = (2pi)^(1/2) (n)^(n+1/2) (e)^(-n), n>0.
