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Probability Density Functions, Transformations and Continuous Random Variables

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Please solve for #3.1.
Please kindly explain each step of your solution.

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PDFs, Transformations and Continuous Random Variables are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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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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