Explore BrainMass
Share

# Joint and Marginal Probability Density Functions of Independent Variables

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

Let X, Y be independent, standard normal random variables, and let U = X + Y and V = X - Y.
(a) Find the joint probability density function of (U, V) and specify its domain.
(b) Find the marginal probability density function of U and V specifying the domain in each case.
(c) Explain why U and V are independent

Joint probability density function of X and Y is given by
f(x,y)=6/7 (x^2+xy/2) 0<x<1. 0<y<2, Find P{X>Y} how to get the answer of 15/56?

https://brainmass.com/statistics/probability-density-function/joint-marginal-probability-density-functions-independent-variables-45665

#### Solution Preview

Hello there

As X and Y are independent, standard normal random variables, their joint pdf f(x,y) is

1/SQRT(2Pi) * Exp[-(x^2)/2]*1/SQRT(2Pi) * Exp[-(y^2)/2] =

1/(2Pi)* Exp[-(x^2+y^2)/2]

As U = X + Y and V = X - Y, we see X = (U+V)/2 and Y = (U-V)/2 so

X^2 + Y^2 = (U^2+ 2UV + V^2 +U^2- 2UV + V^2)/4 = (U^2+ V^2)/2

The probability that (x,y) lies in the incremental area dx dy is given by f(x,y) dx dy

which can be re-written as g(u,v) du dv times the size of the Jacobean ...

#### Solution Summary

The solution shows how to find joint and marginal probability density functions of two independent, standard normal random variables in 355 words.

\$2.19