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# Joint and Marginal Probability Density Functions of Independent Variables

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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

## 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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