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solve part e using the formula from part b

In part **e**, you're asked to find the correlation between \(X\) and \(Y\), and compare the result to the finding from part **b**.

### Formula for Correlation:
The correlation between \(X\) and \(Y\), denoted as \(\rho(X, Y)\), is given by:
\[
\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}
\]
where:
- \(\text{Cov}(X, Y)\) is the covariance between \(X\) and \(Y\),
- \(\sigma_X\) is the standard deviation of \(X\),
- \(\sigma_Y\) is the standard deviation of \(Y\).

### Steps to Compute Correlation:
1. **Covariance \( \text{Cov}(X, Y) \):**  
   The covariance is calculated using:
   \[
   \text{Cov}(X, Y) = E[XY] - E[X]E[Y]
   \]
   From part **b**, you are given a formula for \(E(X^r Y^s)\). You can use this formula to find \(E[XY]\) by setting \(r = 1\) and \(s = 1\).

2. **Means \( E[X] \) and \( E[Y] \):**  
   These are typically computed by integrating the marginal PDFs \(f_X(x)\) and \(f_Y(y)\), which you may have found in part **c**.

3. **Standard Deviations \( \sigma_X \) and \( \sigma_Y \):**  
   Use the variance formulas:
   \[
   \sigma_X = \sqrt{E[X^2] - (E[X])^2}
   \]
   \[
   \sigma_Y = \sqrt{E[Y^2] - (E[Y])^2}
   \]

Once you have \(E[XY]\), \(E[X]\), \(E[Y]\), \(\sigma_X\), and \(\sigma_Y\), you can substitute into the correlation formula to find \(\rho(X, Y)\).

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Do you want me to compute specific values or clarify anything? Here are 5 possible questions that relate to this problem:

1. What is the formula for the marginal PDFs of \(X\) and \(Y\)?
2. How do we compute \(E[X]\) and \(E[Y]\) using marginal PDFs?
3. How is \(E(XY)\) computed directly from the joint PDF?
4. What is the physical interpretation of correlation in this context?
5. How does the independence of \(X\) and \(Y\) affect the correlation?

**Tip:** If \(X\) and \(Y\) are independent, their correlation will be zero. Always check independence to simplify the computation.

Solve part e using the formula from part b

This problem involves calculating the correlation between X and Y from a given joint probability density function (pdf). You will use the covariance formula and expected values derived from part b to solve for the correlation coefficient. Suitable for university-level students studying probability and statistics.

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