Showing Chi squared is independent with another variable

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SUMMARY

The discussion centers on the independence of the random variables Y and Z, where Y is defined as Y = X1^2 + X2^2, resulting in a chi-squared distribution with 2 degrees of freedom. The variable Z is defined as Z = X1 / (X1^2 + X2^2). The conclusion drawn is that Y and Z are not independent, as demonstrated by the relationship Y = Y * Z^2 + X2, indicating that Y depends on Z. The conversation also suggests exploring polar coordinates for further analysis.

PREREQUISITES
  • Understanding of chi-squared distributions, specifically chi-squared with 2 degrees of freedom.
  • Familiarity with independent random variables and their properties.
  • Knowledge of transformations in probability, particularly the use of polar coordinates.
  • Basic algebraic manipulation skills to analyze dependencies between variables.
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  • Research the properties of chi-squared distributions and their applications in statistics.
  • Learn about the Jacobian transformation method for changing variables in probability distributions.
  • Explore the use of polar coordinates in multivariate distributions and their implications.
  • Study examples of dependent and independent random variables to solidify understanding of their relationships.
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Statisticians, data analysts, and students studying probability theory, particularly those interested in the relationships between random variables and their distributions.

torquerotates
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So I have X1 and X2 are iid standard normal.

Then I have Y=X1^2+ X2^2

and

Z=X1/(X1^2+x2^2)

I'm supposed to find the distribution of Y and Z and then determine if they are independent.

Clearly Y is chi squared with degrees of freedom 2.

But I have no idea how to find the distribution of Z. I know there is a shortcut without using the jacobian, like I did with the Chi Squares, but I'm not sure how to do it.
I know Y and Z are not independent because with some algebra,

Y=Y*Z^(2)+X2
So Y depends on Z and vice versa, therefore they cannot be independent. Is that true?
 
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(I moved this thread to the homework section)

I could be interesting to go to polar coordinates (in the X1-X2-plane).

Just finding an equation where Y and Z appear is not sufficient to show a dependence.
Imagine Y=X1, Z=X2, then Y+Z=X1+X2 but they are clearly independent.
 

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