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We are tasked with finding the density of \( R = \frac{X}{X+Y} \), where \( X \) and \( Y \) are independent **Gamma distributed** random variables with parameters \( X \sim \text{Gamma}(\alpha, \lambda) \) and \( Y \sim \text{Gamma}(\beta, \lambda) \).

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### Step 1: Recognize the Problem Setup
The given densities for \( X \) and \( Y \) are:

\[
f_X(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x} \mathbf{I}_{(0, \infty)}(x), \quad f_Y(y) = \frac{\lambda^\beta}{\Gamma(\beta)} y^{\beta-1} e^{-\lambda y} \mathbf{I}_{(0, \infty)}(y).
\]

Both \( X \) and \( Y \) have the same rate parameter \( \lambda \) but shape parameters \( \alpha \) and \( \beta \), respectively. The sum \( T = X + Y \) and the ratio \( R = \frac{X}{X+Y} \) are two key variables here.

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### Step 2: Distribution of \( R \)
The variable \( R \) is known to follow a **Beta distribution** when \( X \sim \text{Gamma}(\alpha, \lambda) \) and \( Y \sim \text{Gamma}(\beta, \lambda) \).

#### Key Property:
If \( X \sim \text{Gamma}(\alpha, \lambda) \) and \( Y \sim \text{Gamma}(\beta, \lambda) \), then:

\[
R = \frac{X}{X + Y} \sim \text{Beta}(\alpha, \beta).
\]

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### Step 3: Beta Distribution Density
The Beta distribution with parameters \( \alpha > 0 \) and \( \beta > 0 \) has the following probability density function:

\[
f_R(r) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)} r^{\alpha-1} (1 - r)^{\beta-1}, \quad \text{for } r \in (0, 1).
\]

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### Final Answer:
The density of \( R = \frac{X}{X+Y} \) is:

\[
f_R(r) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)} r^{\alpha-1} (1 - r)^{\beta-1}, \quad \text{for } r \in (0, 1).
\]

This is the Beta distribution with parameters \( \alpha \) and \( \beta \).

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### Next Steps:
Would you like me to explain why the Beta distribution arises in this context or dive into the derivation? Let me know!

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### Related Questions:
1. What is the intuition behind the Beta distribution in this context?
2. How do we derive the density of \( R \) rigorously using transformations?
3. What happens to the distribution of \( R \) if \( \alpha = \beta \)?
4. How does the sum \( T = X + Y \) relate to the distribution of \( R \)?
5. What is the joint distribution of \( (R, T) \)?

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### Tip:
The Beta distribution often appears when dealing with ratios of independent Gamma-distributed variables. Recognizing such connections can simplify calculations significantly!

What is the density of R = X / (X + Y) where X ~ Gamma(α, λ) and Y ~ Gamma(β, λ), and X and Y are independent?

This problem explores the density of R = X / (X + Y), where X and Y are independent Gamma-distributed random variables. Learn how to derive the Beta distribution, understand Gamma parameters, and see why the Beta density arises in this context.

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