For part c does it matter that X and Y are not independent

No, it does not matter that \(X\) and \(Y\) are not independent when transforming to the new random variables \(U = \frac{X}{Y}\) and \(V = Y\). The transformation process itself relies on the joint distribution of \(X\) and \(Y\), but not on their independence.

### Explanation:

When you perform a transformation of random variables, such as \(U = \frac{X}{Y}\) and \(V = Y\), you are working with the joint distribution of the original variables \(X\) and \(Y\). The new joint distribution of \((U, V)\) is derived using the Jacobian determinant, which accounts for how the probability density function changes under the transformation.

Independence between \(X\) and \(Y\) would imply that the joint pdf can be factored into the product of their marginal pdfs:
\[
f_{X,Y}(x, y) = f_X(x) \cdot f_Y(y)
\]
However, whether or not \(X\) and \(Y\) are independent doesn't affect the validity of the transformation. The resulting joint pdf \(f_{U,V}(u, v)\) is derived from the original joint pdf \(f_{X,Y}(x, y)\), which correctly describes the relationship between \(X\) and \(Y\).

### Key Points:
- The dependence or independence of \(X\) and \(Y\) affects the form of their joint pdf, but the transformation to new variables \(U\) and \(V\) is based on the joint distribution, regardless of independence.
- The joint pdf \(f_{U,V}(u, v)\) that results from the transformation accurately reflects any dependence structure between \(U\) and \(V\) that arises from the original variables \(X\) and \(Y\).

So, while \(X\) and \(Y\) are not independent, this does not impact the process of transforming to \(U\) and \(V\) or the correctness of the derived joint pdf \(f_{U,V}(u, v)\).

Would you like further details on this topic or have any additional questions?

Here are 5 related questions:
1. How would the transformation change if \(X\) and \(Y\) were independent?
2. Can we determine if \(U\) and \(V\) are independent from the joint pdf \(f_{U,V}(u, v)\)?
3. How does the Jacobian determinant affect the transformation of random variables?
4. What are the implications of dependence between random variables when computing conditional expectations?
5. How do you interpret the marginal pdf of a transformed random variable?

**Tip:** When transforming random variables, always ensure to correctly apply the Jacobian determinant, as it accounts for the change in the "density" of probability under the transformation.

Understand how the independence of random variables impacts the transformation to new variables. Explore the relationship between joint distributions, independence, and the Jacobian determinant in mathematical transformations.

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