# Joint and Marginal Probability Density Functions of Independent Variables

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.

Joint and Marginal Distribution Functions Problem

Let X and Y be continuous random variables.

(i) Show that if X and Y are independent, they they are uncorrelated.

(ii) Prove that X + Y and X - Y are uncorrelated if and only if X and Y have the same variance.

Suppose that the joint probability density function of the continuous random variables U and V is given by

f(u, v) = {6e^(-2u-3v), 0,

u >= 0, v >= 0 otherwise

(iii) Show that U and V are independent.

(iv) Find the probability density function of U + V.

(vi) Let P = 2U + 3V and Q = 2U - 3V. Given that the variances of U and V are 1/4 and 1/9 respectively, show that P and Q are uncorrelated.

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