Conditional Expectation, change of variables, change of variables, CharacteristicsFcn


by kliwin
Tags: change of variables, characteristic, conditional, moment generating
kliwin
kliwin is offline
#1
Nov21-12, 01:10 AM
P: 1
1.
Let T = (X,Y,Z) be a Gaussian for which X,Y,Z for which X, Y, Z are standard normals, such that E[XY] = E[YZ] = E[XZ] = 1/2.

A) Calculate the characteristics function Φ_T(u,v,w) of T.
B) Calculate the density of T.

2.
Let X and Y be N(0,1) (standard normals), not necessarily independent. Calculate E[Max(X,Y)] using two different ways:

A) Use the joint density of X and Y and use the fact that for two numbers x and y, max(x,y) = x if x > y and y if y > x.

B) Use the change of variables x = x and y = ρx = u*sqrt(1-ρ^2) in your integral.

3.
Given random variables X and Y whose second moments exit, prove the triangle inequality

E[(X+Y)^2]^(1/2) <= E[X^2]^(1/2) + E[Y^2]^(1/2)

Help guys, this is for my Advanced Probability Course and I am stuck on it. :/
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered

Register to reply

Related Discussions
Conditional expectation of a product of two independent random variables Calculus & Beyond Homework 0
Using change of variables to change PDE to form with no second order derivatives Calculus & Beyond Homework 1
Conditional expectation on multiple variables Set Theory, Logic, Probability, Statistics 2
Expectation conditional on the sum of two random variables Calculus & Beyond Homework 0
3D Change of Variables Calculus & Beyond Homework 2