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Practice Problem: Determine if X and Y are independent
Two continuous random variables X and Y have the following joint probability density function:
$ f_{XY} (x,y) = C e^{\frac{-(4 x^2+ 9 y^2)}{2}}, $
where C is an appropriately chosen constant. Are X and Y independent? Answer yes/no and give a mathematical proof of your answer.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Some students tried this problem on the quiz and try to integrate $ f_{XY}(x,y) $ w.r.t x directly. However, please note that
- $ \int e^{-2 x^2} dx \neq \frac{1}{-2 x^2} e^{-2 x^2} $
Answer 1
Hint:
- X and Y are independent iff $ f_{XY}(x,y)= f_{X}(x)f_{Y}(y) $
- $ f_{X}(x)= \int_{-\infty}^{\infty} f_{XY}(x,y)dy $
- When integrating w.r.t. y, x can be viewed as a constant and thus you can pull the term associated with x outside the integral.
- Try to reformulate the integrand to a Gaussian pdf with a coefficient.
- Use the property that the integration of Gaussian pdf equals 1.
Answer 2
Write it here.
Answer 3
Write it here.
