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 $\rho(X, Y)$.

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.