To derive the expression:
$\frac{\text{cov}(R, G)}{E(R)} = (\text{corr}(R, G)) \cdot \sqrt{\frac{1 - E(R)}{E(R)}} \cdot \text{var}(G),$ let's break it down step-by-step:

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Definitions and Formulas

1. Covariance ($\text{cov}(R, G)$): $\text{cov}(R, G) = \text{corr}(R, G) \cdot \sigma_R \cdot \sigma_G,$ where:

   • $\text{corr}(R, G)$ is the correlation coefficient,
   • $\sigma_R$ and $\sigma_G$ are the standard deviations of $R$ and $G$, respectively.

2. Variance ($\text{var}(G)$): $\text{var}(G) = \sigma_G^2.$

3. Standard deviation of $R$: $\sigma_R = \sqrt{\text{var}(R)}.$

4. By definition of expected value: $E(R) = 1 - \text{var}(R) \quad \Rightarrow \quad \text{var}(R) = 1 - E(R).$ This assumes $R$ is constrained between 0 and 1 (common in probability).

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Step 1: Start with $\frac{\text{cov}(R, G)}{E(R)}$

Substitute the formula for covariance: $\frac{\text{cov}(R, G)}{E(R)} = \frac{\text{corr}(R, G) \cdot \sigma_R \cdot \sigma_G}{E(R)}.$

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Step 2: Express $\sigma_R$ in terms of $E(R)$

From the variance formula: $\text{var}(R) = 1 - E(R).$ So, $\sigma_R = \sqrt{\text{var}(R)} = \sqrt{1 - E(R)}.$

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Step 3: Substitute $\sigma_R$ and $\sigma_G^2 = \text{var}(G)$

$\frac{\text{cov}(R, G)}{E(R)} = \frac{\text{corr}(R, G) \cdot \sqrt{1 - E(R)} \cdot \sqrt{\text{var}(G)}}{E(R)}.$

Rewriting $\sqrt{\text{var}(G)}$ as $\sigma_G$, we get: $\frac{\text{cov}(R, G)}{E(R)} = \text{corr}(R, G) \cdot \sqrt{\frac{1 - E(R)}{E(R)}} \cdot \text{var}(G).$

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Final Result

Thus, the expression is verified: $\frac{\text{cov}(R, G)}{E(R)} = (\text{corr}(R, G)) \cdot \sqrt{\frac{1 - E(R)}{E(R)}} \cdot \text{var}(G).$

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Would you like further clarification or examples? 😊

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Related Questions

1. What is the significance of covariance and correlation in probability?
2. How does the relationship $E(R) = 1 - \text{var}(R)$ arise?
3. How can you interpret $\text{var}(G)$ and $\text{corr}(R, G)$ in this context?
4. What happens when $E(R) = 0.5$ in the formula?
5. Can you explain how this formula applies to real-world problems, such as financial returns?

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Tip

When dealing with expected values and variances, always check the range of your random variables to ensure consistency in assumptions.