1. **Stating the problem:** We are given a function of the form $$f(x|r) = C(r)(1-x)^{a(r)-1} x^{b(r)-1}$$ where $$r > 0$$ and $$x \in (0,1)$$. 2. **Understanding the parameters:** The parameters satisfy $$0 < a(r), b(r), C(r)$$ and $$a(r) = a + b + r$$ for some constants $$a, b > 0$$. 3. **Interpreting the function:** This function resembles a Beta distribution probability density function (PDF) with shape parameters $$a(r)$$ and $$b(r)$$, scaled by a constant $$C(r)$$. 4. **Formula for Beta PDF:** The Beta PDF is given by $$ f(x) = \frac{x^{\alpha - 1} (1-x)^{\beta - 1}}{B(\alpha, \beta)} $$ where $$B(\alpha, \beta)$$ is the Beta function. 5. **Matching terms:** Here, $$\alpha = b(r)$$ and $$\beta = a(r)$$, so $$ f(x|r) = C(r) (1-x)^{a(r)-1} x^{b(r)-1} $$ To be a valid PDF, $$C(r)$$ must be the reciprocal of the Beta function: $$ C(r) = \frac{1}{B(b(r), a(r))} $$ 6. **Behavior of the function:** Since $$a(r) = a + b + r$$ and $$r > 0$$, $$a(r)$$ increases with $$r$$, making the function more skewed depending on $$r$$. 7. **Summary:** The function $$f(x|r)$$ is a Beta distribution PDF with parameters $$\alpha = b(r)$$ and $$\beta = a(r) = a + b + r$$, defined on $$x \in (0,1)$$. **Final answer:** $$ f(x|r) = \frac{x^{b(r)-1} (1-x)^{a + b + r - 1}}{B(b(r), a + b + r)} $$ where $$B(\cdot, \cdot)$$ is the Beta function.